The trouble with inventing imaginary countries is that eventually someone will insist on believing they're real. For instance, after Sir Thomas More published his utopian tract Utopia in 1516, Guillaume Budé wrote to him to urge that missionaries be sent there to convert the natives to Christianity. The all-time record in this respect is held by Plato's lost continent of Atlantis, which has been believed in more firmly by more people than any other imaginary country in history.
Around 360 BC, the philosopher Plato decided to follow up his hit Socratic dialogue The Republic with a trilogy of philosophical blockbusters: Timaeus, Critias, and Hermocrates. In order to demonstrate how the ideal state described in The Republic would work in real life, Plato had the character Critias mention an Egyptian legend about a continent beyond the Strait of Gibraltar called Atlantis that had once had an ideal society just like the one described in The Republic. Sadly, Atlantis got into a war with another ideal society located in Athens, and the war ended when Atlantis sank into the sea. That was the outline Critias gave in Timaeus, and Plato intended to have him go into greater detail in his own dialogue. As it happened, though, Plato gave up on his projected trilogy halfway through Critias, and the full story of Atlantis was never told.
Plato, needless to say, didn't expect anyone to actually believe that Atlantis had really existed, and for several centuries after Timaeus came out, no one did. After the rise of Christianity, however, gullibility became fashionable, and people started to believe that Atlantis was real, and they've never stopped. Since then, Atlantis has been joined by a couple of other imaginary-lost-continents-that-people-think-were-real called Mu and Lemuria, but Atlantis still has place of pride (you won't, for instance, hear of any space shuttles being named Mu).
Countless writers, visionaries, and cranks have maintained Plato's tradition by peopling their versions of Atlantis with their own ideal societies. In his 1954 book Lost Continents, L. Sprague de Camp stated that nearly two thousand books and articles had been written about Atlantis and other lost continents, and the pace has not slackened since then. Atlantis has appeared in the 1975 Illuminatus Trilogy, at least three episodes of Doctor Who, and (of course) the Stargate sequel series Stargate: Atlantis.
Up until the late 19th century, though, arguing that Plato's Atlantis was a real place was an intellectual exercise indulged in by the occasional pedant. What made Atlantism a popular craze was the publication of Atlantis: The Antediluvian World by Ignatius Donnelly in 1882. Donnelly popularized the idea that Atlantis was the original source of human civilization, and that later civilizations that appeared in Central America and Egypt were established by Atlantian colonists before their own land drowned.
The Atlantis craze was also helped along by a proposed solution to a scientific puzzle. Back in the 19th century, paleontologists noticed a curious thing about the fossil records of Europe and North America: up until the end of the Triassic period, the same species evolved at the same time on both continents, but during the Jurassic period different species evolved on each continent. Geologists assumed that this meant that a land bridge connected the two continents up until the Triassic period, and then sank during the Jurassic period.
Atlantists seized on this idea as proof that huge land masses could indeed sink into the sea, and this gave a boost to Donnelly's Atlantis-as-the-source-of-all-civilization idea, a boost that was still going on in the 1930s when Amelia Reynolds Long decided to use it for her story "Reverse Philogeny". Using the idea of a "race memory" (another popular bit of pseudoscience at the time), she wrote her second Professor O'Flannigan story, telling how O'Flannigan used hypnosis to recover race memories of the fall of Atlantis. As many Atlantists did, Long improved on Plato's story by having the Atlanteans possess 20th century technology -- in this case, automobiles. Long also echoed contemporary Atlantists (who echoed Donnelly) by having Europe be an Atlantean colony, so that Europeans (and their American cousins) could claim to be heirs to the world's oldest civilization -- so take that, you snotty Asians, with your Mesopotamia!
(Incidentally, another manifestation of Atlantism in the 1930s occurred in Oxford, when the philologist J. R. R. Tolkien added a drowned continent motif to his Silmarillion story cycle. The literal rise and fall of the island continent of Numenor eventually became the Second Age of Middle-earth in the back-story to The Lord of the Rings.)
When the theory of plate tectonics was established in the 1960s, and geologists recognized that continental drift had actually happened, the land bridge theory went out the window, and with it the Atlantists' chief claim to scientific confirmation of their beliefs. Continents do not rise and fall thousands of feet, and Plato did, in fact, create Atlantis out of whole cloth -- as his student Aristotle put it, "he who invented it also destroyed it". Since then, no serious person has suggested that Atlantis ever existed (though countless unserious people continue to do so).