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When we say "high strangeness" in UFO cases, we are talking about the bizarre elements on top of what is already a strange event. Some of them happen in the aftermath, some during the incident. Here is an example:

August 6, 1977, near Pelham, Georgia (20 miles north of Thomasville), in Mitchell County. At 10:30 a.m., retired automobile salesman Tom Dawson (sixty-three years old, at the time) took a walk down to his favorite pond to see how it looked for fishing later that day.

Just as he got inside the fence surrounding the pond, a circular space ship zipped right in between the trees and hovered just a few feet above the ground. At the same time he found himself, his two dogs and twenty head of cattle, frozen in place by an unseen force.

Dawson said the craft was about 15 feet high and 50 feet in diameter. It had portholes all around and a dome on top. It made no sound and changed colors rapidly from one to another. Suddenly, a ramp came down and out came seven hairless, snow-white beings, about 5 feet tall, with pointed ears and noses.

Some had on tight-fitting one-piece suits while others wore nothing. They talked in a 'high-pitched' gibberish he could not understand.

They conducted what he thought to be a medical exam of some kind. They placed a skullcap-like device on his head and a large hula hoop-shaped thing (connected to a box) around his midsection. After they had collected "some leaves and stuff," they got back on the ship and were gone in the blink of an eye.

Once free, Mr. Dawson ran uphill (about 300 yards) to his trailer. He was having trouble breathing and talking, so he was taken to the Mitchell County Hospital, where the doctor said he had been shaken both mentally and physically from his encounter with the UFO and its occupants. He was treated for hysteria (given something to calm him down) and later released. Dawson said he believed that if he had been a younger man the extraterrestrials would have taken him away. - Billy J. Rachels / Billy Norris - UFO Bureau of Georgia

Above is the basic version of "the Dawson encounter" of 1977, below is the version which adds the high strangeness elements:

Location: Pelham, Georgia
Date/Time: August 6, 1977 - 10:30 am local time

"I AM JIMMY HOFFA"

Tom Dawson, 63 and retired, was walking with his two dogs from his trailer home to a nearby farm when a "circular shaped space ship" with a dome and portholes, 40-50 ft in diameter and 12-14 ft high, hovered several feet off the ground in a field in front of him. He immediately noticed that he was unable to move a muscle as well as his dogs and the cattle in the field seemed likewise to be immobilized.

A hatch in the object opened, a ramp was extended, and 7 strange looking humanoids 5 ft tall emerged, five men and two women. The first stepped down cautiously, as if to test the solidity of the ground; then the rest followed, with two taking up sentry positions at the hatch. They were all hairless, with skin "as white as a flour sack," their noses were sharp and turned up, their ears were pointed, and they had no necks. One of the men and one of the women were completely nude. The clothing of the rest, male and female alike, was very beautiful with silk like shoes with pointed, turned up toes.

The humanoids cautiously approached Dawson and gave him a kind of physical examination, placing on his head a skullcap with cords connecting it to a hoop bearing dials. They dropped his trousers and lifted his shirt for the examination, passing the hoop over his body. While the examination was in progress, a loud voice came from the object, shouting three times "I am Jimmy Hoffa" (!) a fourth repetition was cut off, and the voice was not heard again.

After completing the examination, all returned to the craft except two men, who walked about 10 feet away and "went into a conference." They had very shrill voices, and although Dawson was unable to comprehend what it was they said, he thought he heard the word "Jupiter." He had the impression they were debating whether or not to take him on board; at any rate, they did not, and the leader passed his palm across his chest as though to signal good-bye. They reentered the ship, closed the hatch, and took off. He saw the object rise to 75 feet then, in a wink, it was out of sight. Dawson was then freed of his paralysis; he ran directly to his next-door neighbor, but was so excited he could say nothing more than "space ship!" He was taken to a hospital and treated for hysteria. - Billy J. Rachels - UFO Bureau of Georgia (Source for both versions.)

Notice the stuff that is cut out out of the first version: the naked "aliens" could be easily distinguished as male and female by their human-like sex organs, that the creatures had no necks, and the high strangeness element of the voice shouting "I am Jimmy Hoffa." At the time of the incident, James Hoffa had been missing since July 30th, 1975 and would not be declared dead until the early 1980s. The element of temporary paralysis is common; a good example would be the Valensole Case of 1965, where French lavender farmer Maurice Masse was zapped by two "aliens" who had landed their small "ship" in his field; he was stuck in place for 20 minutes. One of the creatures pointed a small tube at Masse to produce the effect.

I've read about similar cases. There's too many. I'll post some detailed famous cases in a few months when I have time, although UFOs is not a topic admittiedly in my specialty.

Because I don't have another UFO thread here is Alexander Wendt (Ohio State University) and Raymond Duvall's (University of Minnesota) 2008 political science paper "Soverignty and the UFO" in PDF format, which is a great explanation for why governments around the world have been avoiding doing real research on the topic - if there is a Higher Power (extraterrestrial or ultra-dimensional) involved with subject, then the State as highest form of political power is nullified. Good rips on the skeptics movement as well:

“It Can’t Be True”

Given the inconclusiveness of the empirical record, UFO skepticism ultimately rests on an a priori theoretical conviction that ET visitation is impossible: “It can’t be true, therefore it isn’t.” Skeptics offer four main arguments to this effect.

“We are alone.” Philosophers have long debated whether life exists beyond Earth, but the debate has lately intensified in response to empirical discoveries like extra-solar planets, water on Mars, and “extremophile” organisms back home. A thriving discipline of astrobiology has emerged, and the view that life exists elsewhere seems poised to become scientific orthodoxy.

However, this does not mean that (what humans consider) intelligent life exists. The only evidence of that, human beings, proves merely that intelligence like ours is possible, not probable. The Darwinian “Rare Earth hypothesis” holds that because evolution is a contingent process, human intelligence is a random accident, and the chances of finding it elsewhere
are therefore essentially zero.

This is a serious argument, but there is a serious argument on the other side too, going on within evolutionary theory itself, where the neo-Darwinian orthodoxy is today being challenged by complexity theorists. Rather than contingency and randomness, complexity theory highlights processes of self-organization in Nature which tend toward more complex organisms. If the “law of increasing complexity” is correct then intelligent life might actually be common in the universe. Either way, today it is simply not known.

“They can’t get here.” Even if intelligent life is common, skeptics argue it is too far away to get here. Relativity theory says nothing can travel faster than the speed of light (186,000 miles per second). Lower speeds impose a temporal constraint on ET visitation: at .001 percent of light speed, or 66,960 miles per hour—already far beyond current human capabilities—it would take 4,500 Earth years for ETs to arrive from the nearest star. Higher speeds, in turn, impose a cost and energy constraint: to approximate light speed a spaceship would need to use more energy than is presently consumed in an entire year on Earth.

Physical constraints on inter-stellar travel are often seen as the ultimate reason to reject the ETH, but are they decisive? Computer simulations suggest that even at speeds well below light the colonization wave-fronts of any expanding ET civilizations should have reached Earth long ago. How long ago depends on what assumptions are made, but even pessimistic ones yield ET encounters with Earth within 100 million years, barely a blip in cosmic terms. In short, ETs should be here, which prompts the famous “Fermi Paradox,” “Where are They?”

Additionally, there are growing, if still highly speculative, doubts that the speed of light is truly an absolute barrier.
Wormholes—themselves predicted by relativity theory—are tunnels through space-time that would immensely shorten the distances between stars. And then there is the possibility of “warp drive,” or engineering the vacuum around a spaceship, enabling it to skip over space without time dilation. Speculative as these ideas are, their scientific basis is sufficiently sound that research is currently being funded through the “Breakthrough Propulsion Program” at NASA. They may prove to be wrong or beyond human capacity. But if humans are imagining them just 300 years from our scientific revolution, what might ETs 3,000 years, much less 3,000,000, from theirs be imagining?

Most important graph from that section: .....Given the stakes, ignoring UFOs only makes sense if human beings can
be certain they are not ETs. We have shown there is more than reasonable doubt: the ETH cannot be rejected without significant risk of Type II error [ i.e., failing to reject a false null hypothesis, S.]. What is actually known about UFOs is that we have no idea what they are, including whether they are alien; far from proving UFO skepticism, science proves its ignorance. With so little science on either side, therefore, the UFO controversy has been essentially theological, pitting ET believers against unbelievers. In this fight, the unbelievers have secured the authority of science, giving them decisive advantage. [bolding by S.] Their views are taken as fact, while those of believers and agnostics are dismissed as irrational belief. Since science does not actually justify rejecting the ETH, why would unbelief be so hegemonic? The UFO taboo is puzzling, we submit, and demands a deeper look at how its “knowledge” is produced.

A great chronological list of every "Men in Black" case ever reported from 1924 to 2003, Magonia Review's Men in Black Encounters, a Short Catalogue. A sample case:

1967, Christmas season: Peggy G., who had had two UFO sightings earlier that year, and later had a poltergeist in her home, worked in a department store in the afternoon. A guard employed there asked her to lunch, and “informed that he was a member of a secret organisation working on earth called the ‘Cosmic Brotherhood’”. When a co-worker mumbled that he was crazy the guard looked up and told him to get away, rays of light shooting from his eyes. Soon after he left his job and could not be traced. Later she had interference on her telephone, and saw two men stringing silver tape over the wires near her home. The police, when called, commented: “Oh, the silver tape again.” Beckley, The UFO Silencers, pp.17-19.

I copypasted this one because it has elements that are outside of "normal" UFO cases (the claimed poltergeist) while the Men in Black (a term coined by John Keel, author of The Mothman Prophecies) in the case are the "Cosmic Brotherhood" guard and the guys stringing the silver tape on the telephone lines.

1947. 22 June: Harold Dahl was visited at 7 a.m. by a man dressed in black, who drove him in a black Buick sedan to a café where he told him about his sighting of six ‘doughnut’ shaped objects the day before near to Tacoma, Washington State, in such detail that he could have been there; and said that if “he loved his family he would keep quiet about the matter.” Wilkins, Flying Saucers on the Attack, pp.51-62 ; Randles, MIB, pp.30-31; and several others. Dahl was later questioned by two Air Force intelligence officers, Frank Brown and William Davidson; when they set off by air to return to their base, the plane crashed and they were killed. Two days later Kenneth Arnold, who had also investigated the affair, was flying home when his engine cut out and he was forced to crashland. It has become common for writers to say that Dahl admitted that the story was a hoax, but an August 1947 teletype from the Seattle FBI Special Agent George Wilson to J. Edgar Hoover stated that: “Please be advised that Dahl did not admit to Brown that his story was a hoax but only stated that if questioned by authorities he was going to say it was a hoax because he did not want any further trouble over the matter.” Keith, Casebook, p.46.

This is where you start hearing that the "men", when they pose as government special agents or act like they are such people, wear black suits or other black clothing, and that they drive black cars, many times with plate numbers that have never been issued or driving without license plates at all.

1955: Twenty workmen were repairing the outside of a factory in southern New Jersey, which was engaged in classified work for the navy, when they saw a gigantic circular object descend and hover over the car park for several minutes. As they were about to clock out, a man in civilian clothes herded them into a meeting room, where he flourished a sheaf of papers, saying: “We want you all to sign an oath of secrecy promising not to tell about what you saw today. Those of you who don’t want to sign needn’t come in to work tomorrow – or ever again.” Everyone signed. John Keel stated that this story “is more folklore than fact. The story has circulated by word of mouth for years, but no one has even pinned down any of the original witnesses, if they exist.” Keel, The Cosmic Question, pp.151-52.

1957, November: Olden Moore watched a circular machine land near Montville, Ohio, on 6th, a few days later the local sheriff drove to his house with men in air force uniforms, they took him to the field where he had seen the UFO, a helicopter was waiting there, he was flown to an airport and put on a plane to Washington, where he was imprisoned for three days, and two officers tried to get him to admit he had seen nothing but a ‘fireball’. Finally he was flown back to Ohio, but later neither the sheriff nor the Air Force would back his story. Keel, The Cosmic Question, pp.155-56.

Above are two examples where the MIBs definitely are human and working for civilian or military intelligence. When threats are made, they seem to always be nasty whether or not the men in black are human or "other" and any documents or photographs handed over to them are never returned.

Strelnikov wrote:A great chronological list of every "Men in Black" case ever reported from 1924 to 2003, Magonia Review's Men in Black Encounters, a Short Catalogue.

How enlightening that the reports peaked out in the late 1960s and declined afterwards.

ericbarbour wrote:

Strelnikov wrote:A great chronological list of every "Men in Black" case ever reported from 1924 to 2003, Magonia Review's Men in Black Encounters, a Short Catalogue.

How enlightening that the reports peaked out in the late 1960s and declined afterwards.

Actually their list ends in the early 2000s - I have a report here of an encounter in Australia in 2008 that has all the hallmarks of classic Men in Black-ery: the character gave a BS cover story to the witness to demand some recorded evidence (cellphone video in witnesses' mobile phone) of a UFO the witness had seen, they used a hypnotic effect to keep the witness from running away while they snatched the cellphone away, they wore out-of-season clothes (long dark trenchcoat, black beret), they showed up out of the blue a few days after the sighting when the witness (as far as I know) did not report his sighting to the police or media.

The MiBs are now part of strange phenomena - every time there is a witness, there will be reports of "silencers" trying to get evidence or telling that witness not to speak to investigators or the media.