Experimental Archeology

How do you prove your controversial theory about a feat that seems almost mythic in quality, despite the fact that every expert in the field in question says it's impossible? By doing it yourself! All you have to do is recreate the situation and prove that it really was possible.

One common variant of this involves proving that a voyage in an ancient story really happened. This tends to involve building a replica of the vehicle in question as closely as possible (although modern creature comforts often tend to make their way into the finished work), finding a crew as loopy as you are, and riding your way to vindication. Provided, of course, you live to tell the tale.

Adventurer Archaeologist characters may be prone to trying out this method.

Anime and Manga

 * One of these plots becomes the center of One Piece's Skypeia saga, where the crew, almost as an afterthought, seeks to prove that the hero of an old fairy tale was not a liar..

Comic Books

 * In a Silver Age Jimmy Olsen story, Perry White leads the the Daily Planet staff on a dangerous trek across the desert in order to prove that an ancestor his really did save the lives of a group of soldiers and settlers by doing just that.
 * In one Donald Duck story, Donald gets involved in a bet to discover which means did the ancient Duckburgers use to immigrate from a local Easter Island stand-in.

Live Action TV

 * In the Star Trek: Deep Space Nine episode "Explorers," Ben and Jake Sisko sail to Cardassia in a Bajoran lightship that travels on solar winds to show a supposed ancient voyage was possible. In a small twist, the Cardassians scoff at the idea that Bajorans were able to make that journey, but as soon as the Siskos were able to do it themselves, the Cardassians apparently decided to reveal that they "coincidentally" (read: kept under wraps until now) found the remains of an ancient Bajoran lightship to share in the glory of Sisko's feat by providing proof that the ancient voyage actually happened.
 * Which was one of the finer moments in Gul Dukat's 'anti-hero' period; he's not entirely pleased that Sisko embarassed the Cardassian people by accomplishing that which they maintained was impossible, but he's gracious enough. Thanks to Marc Alaimo's amazing ability to handle Dukat's complex characterization, particularly attentive viewers will even detect a subtext of honest sincerity and admiration in his 'congratulations' speech.
 * Pretty much the entire point of MythBusters, although fairly rarely regarding actual voyages, with notable instances of such being the reenactment of the famous Alcatraz Escape (Turns out, you can (though it would take a bit of luck and almost-ideal conditions)), and seeing if it was possible to launch a Chinese astronomer into orbit with regular rockets (Turns out you can't).
 * While never used for an actual voyage, they did try making a boat out of newspaper and ice, based on a material developed during WWII to build ships quickly and cheaply.
 * Parodied in Monty Python's Flying Circus: "Mr. and Mrs. Brian Norris's Ford Popular" tests a theory that the population of Hounslow (a suburb of London) originally emigrated from Surbiton (another London suburb). The sketch actually mentions the Kon Tiki (see the Real Life section below) in the beginning.
 * The BBC documentary series Around The World in Eighty Days had Michael Palin travel around the world in eighty days much in the same manner as the original novel's Phileas Fogg (i.e. no air travel). He had certain advantages like better transportation infrastructure, but the series was filmed several years before The Great Politics Mess-Up with all that that entails, so it probably evens out.

Literature

 * Phileas Fogg does this in Around the World In 80 Days, likely making Verne the Trope Maker, since the book is Older Than Radio.

Western Animation

 * Tiny Toon Adventures parodied Heyerdahl's Kon-Tiki expedition in "Voyage of the Kon-Ducki," in which Plucky Duck proves that his ancestors sailed to Salinas, California in the 1970s (instead of flying).

Real Life

 * The most famous example of this (and the former Trope Namer) is probably the Kon-Tiki expedition by Thor Heyerdahl, who sailed an Incan balsa raft from Peru to the Tuamotu Islands in 1947 to demonstrate the plausibility of South American contact with Polynesia. Ironically (with regards to the typical outcome of such stories as invoke this trope), archaeologists have since decided that Polynesians weren't Incan sailors after all and that Heyerdal's theory was wrong. Even at the time, Heyerdal was criticized for being towed out several miles past the shore before sailing... which skipped some extremely dangerous tides that would have made sailing near-impossible.
 * Besides the Kon-Tiki expedition itself, Thor Heyerdahl also tried sailing across the Atlantic ocean using only ancient Egyptian boatmaking technology - essentially, grass. The grass boat Ra II made it within a few miles of the South American coast before being sunk by a storm.
 * Heinrich Schliemann tested a possible candidate for the historical Troy by running around the city walls, based on a passage in The Iliad.
 * National Geographic is fond of doing this, whether it's recreating Odysseus' voyage or trying to build a pyramid using only technology available to Ancient Egyptians.
 * In 1916, the Smithsonian Institution claimed the "Aerodrome," invented by Sameuel P. Langley, had been the first manned heavier-than-air aircraft based on a recreation of the Aerodrome by Glenn Curtiss. The recreation was claimed to have been accurate, which had the Smithsonian claiming that the Wright Brothers flyer to tertiary status (the first one was Santos Dummont). The Smithsonian didn't correct this until 1942. The Aerodrome flown by Curtiss had been heavily modified and was not true to the original design.
 * It's generally popular with modern historians to lend support to a particular theory of how apparently unlikely things occurred in the past. There was a crew that moved a Stonehenge-size piece of stone from somewhere near Germany to location of Stonehenge without any modern equipment and another guy in England who made a Stonehenge piece and then tilted it upright and planted it single-handedly.