Any event of the Society For Creative Anachronism, where you can see examples of everything from 16th-century German armor to Moors to Vikings to Romans to Samurai all competing in the same tournament, or a Viking chatting with an Elizabethan lady over a display of Catholic prayer beads. Largely a result of the basic structure of the organization; anything before 1600 AD is fair game for re-enactment. Individuals will generally be faithful to a period in their dress, though this also varies.
The unofficial motto of the SCA is: "The Middle Ages... as they should have been."
Medieval Re-enactors in the UK used to annually attend the Battle of Tewkesbury each year, but until recently, it was well-known for allowing anyone with vaguely medieval kit (garb to SCA types) onto the field. At one of the last events before it changed to a more authenticity based re-enactment, it wasn't uncommon to see 9th century Vikings, the (ubiquitous) kilted Scotsmen and Normans from 1066 fighting against bill blocks.
Renaissance festivals in general are often actually more of a pastiche of Middle Ages, Renaissance, and Fantasy (including fairies, etc.).
Sci-Fi, what with the occasional incursions of groups of people in Star Trek uniforms pretending to be an Away Team.
Or the obligatory Imperial Stormtrooper in a kilt, with a sword on his hip.
Camlann Medieval Village, outside Seattle, does an admirable job of being as authentic as possible. Unfortunately, drive for authenticity doesn't exempt them from liquor laws, fire codes, or health regulations — forcing them to institute smoke detectors, ID checks, and plastic take home containers.
Cowboy Action Shooters love this trope. While costumers are encouraged to wear appropriate costume, there is no rule forbidding (for example) a Confederate soldier from using an M1887 Winchester shotgun. Part of this is because the governing organization permits and encourages Western movies to be used as inspiration, and classic Westerns are notorious for this trope.
During the 2008 campaign, then-Senator Joe Biden said "Now, when this country entered the Great Depression, our president, Franklin Roosevelt, went on television and spoke of how to get this country out of it." The Daily Show had a lot of fun with this, since 1) Herbert Hoover, not Roosevelt was President at the time, and 2) Television barely existed at that time.
Hoover did appear on an early television broadcast, though. And yes, this was either while or just before he was President, not at some point before he died in 1964.
Al Gore stated in an interview that he grew up reading The Very Hungry Caterpillar as a kid. The book was written and first published in 1970, a year after Gore graduated from university.
Modern printings of Old English works gleefully use the letter 'W', which would not be created for a few hundred years; however, the only other option would be to use the contemporary letter wynn ('Ƿ'), which was abandoned due to constant confusion of it and 'P'.
On a related note, a lot of people think that there used to be an English word 'ye', as in the faux-historical phrase "Ye Olde Shoppe". This is actually a variant form of the character thorn (Þ, unicode U+00FE), which in the script form resembles a 'y' with either an accent over it or, as is commonly rendered on faux-archaic signage, with horizontal lines leading off to the sides from the ends of the upper arms.
People taken in by this trope often turn up at antiques markets or pawnshops, honestly believing that some dusty old item they found in the attic is much older (or occasionally much newer) than it actually is.
People's memories can be this way. Some people being interviewed about their experiences surviving the great Johnstown Flood of 1889 years after it happened mentioned that they checked their wristwatch - even though wristwatches didn't become popular until the 1920's, 30 years later. Their memories could have melded it with the similar (but less well-known) Johnstown Flood of 1936.
Sam Wineburg's Historical Thinking and Other Unnatural Acts argues that most of us see time as a sort of dual choice: the present and "the past," which includes everything from cavemen to toga-wearing Romans to medieval kings and knights to Elvis Presley, and that it's really a fairly unnatural (and therefore difficult) act to periodize the past and think of it as a flowing narrative rather than a list of events. In fact, most people up until relatively (in a historical sense) recently wouldn't think of history any other way. If you told that Medieval artist in the opening of this topic that he was wrong in painting Jesus in Medieval Italian attire, he would just scratch his head and look at you with a puzzled expression.
As mentioned on the Real Life page, most people's lives invoke this to some extent as different people have different incomes and priorities, and the service life of much modern technology reaches deep into obsolescence. So, any given person might own the very latest desktop computer but still be using a Nokia "Brick", or they might have an iPhone 4S and a '92 Civic, for example. Or they might be using a microwave made in 1984 to cook something they prepared using the latest kitchen gadget.
This happened, after a fashion, at various times in history when much more technologically advanced civilizations came into contact with isolated and more rural civilizations. Europeans settling the Americas, the late Edo and early Meiji Periods of Japan, WWII cargo cults, and so on. The previously isolated civilization incorporates technology that is effectively decades or even centuries ahead of what they had before into their lifestyle.
This still happens as parts of the world that are considered "third world" skip several generations of technological advancement. Cell/smartphones are the quintessential example: instead of stringing copper to do landlines first, many places without any telephone service now simply start with wireless, this leading to scenes where a Masai cattle herder, otherwise looking very traditionally dressed, hauls out his Nokia to answer a text.
This can be seen in America with the Amish who use some types of modern technologies alongside a mostly agrarian lifestyle more typical of the early 20th century than the early 21st.
People who engage in "traditional activities" frequently exhibit this trope. In the Canadian Arctic, one can still find people who use a dog team to go hunting. Use of the dog team goes back thousands of years, the sled probably uses metal and plastic, the rifle might be a century-old design, and there's good odds they're carrying a GPS and/or satellite phone.
Though in those cases it's often because the "traditional" millenia-old way of doing things still has a lot of advantages over the newer methods. A dog sled is better suited to snowy terrain than most vehicles and the acute senses and natural pathfinding abilities of the sled dogs can be invaluable. There are documented cases of sled dogs instinctively dragging their master back to civilization after he became sick or injured, saving his life.
During World War I, trench warfare and rapid-firing machine guns brought back many artifacts of war that had been rendered obsolete decades or centuries before, such as metal helmets (the Commonwealth forces based their common soldiers' on the ones of English Medieval archers; machine gun crews adopted others closer to knights), maces, hand grenades, pavise-style shields (for sharpshooters) and chain mail and plate armor for assault troops. The advent of bomber planes and tanks made trench warfare obsolete in turn, and put an end in consequence to most of these 'revivals'.
World War One was riddled with this. Heavy artillery was often pulled by oxen. Tanks, machine guns, and planes all existed but cannons were still used on front lines sometimes, and older tactics like straight up infantry advances (which usually just resulted in a very high body count) were still in use.
What's more, you also had modern democratic nation-states (like the US) fighting with and against centuries-old empires and absolute monarchies ruled by Kings and Emperors. No such monarchies exist in Europe today; most monarchs now are just figure-heads for democracies.
The Pre-Columbian American cultures, developed in complete isolation from Eurasian cultures, are often this when looked at from a Eurasian-centric perspective. People in Mesoamerica at the beginning of the 16th century lived in cities with step pyramid temples like the ones in Ancient Babylon, but had a near total literacy rate, which in most of the world was not seen until the 20th century. They made their weapons out of stone but they had gold metallurgy (the exception being the Tarascans that had bronze weapons) and used propulsors (atlatls) in warfare, that are more associated with Paleolithic hunters in Europe. They had no draft animals and so ignored the wheel, along with the potter's wheel and the windmill, but the Mayas built wheeled toys for children. And so on.
Meanwhile, the Incas in South America had pack animals (llamas) but no wheel nor carts, yet they built an extensive net of roads and forts to facilitate their conquests like the Romans did in the Old World. Their military organization also resembled Old World cultures (as noted by the Spanish that compared them to the Turks), displaying spearmen on Greek phalanx-like formation and using weapons, helmets and pectorals of bronze. The halberd, a weapon invented in Eurasia to fight armoured knights that did not exist in any shape in the Americas since horses went extinct there before they could be domesticated, was common. Their ships resemble an odd mix between Egyptian skiffs and Viking longboats, and are believed capable of sailing to Polynesia and Mexico. And yet, despite this well-built state structure the Incas never had writing: laws and history were memorized by people tasked with it, and math operations were done tying knots in kipus.
Some tribal societies in Africa rely mostly on their traditional ways, living in traditional huts and wearing traditional clothes, but can be seen carrying around AK-47s and other modern weaponry for defense in war-torn areas. In a similar vein, many foreign aid programs send unwanted clothing from American and Canadian thrift shops such as Savers to impoverished people in Asia and Africa, so it's common to see photos of people farming or living in mud-floor homes while wearing shirts with glitter-text English slogans on them.
This trope is the reason given in thisReddit thread for why women find the "fedora persona" creepy and offputting rather than charming. It's not that it's not nice to be a gentleman; it's that it's jarringly unnatural to act like every Hollywood History interpretation of the 'gentleman' from between 1750 and 1950 at once.