Director Danny Boyle addressed that question in an interview that appears on Variety's Web site. Apparently the answer is it's based on rumors that inspired a novel.
Here's the relevant passage:
Variety: How true is the story that this is based on, and how do you find that out?
Danny Boyle: Well, there's a lot of realism in the story. It's quite difficult to pin things down in India sometimes, particularly in Mumbai, which is the home of the Bollywood film indsutry so exaggeration is ever-present. There were rumors about this story, and there is some truth in it. And there are certainly some examples of highly educated professionals like lawyers and teachers and professors disguising themselves as rickshaw drivers and going on the show [but] the show supposedly tweaked this that this was going on quite early on, so the questions are actually the most difficult questions [on any "Who Wants to Be A Millionaire" program] in the world. They're much more difficult than the American version. After you get over the first couple which are easy, to settle you in, they jump to kind of nuclear physics level almost immediately. They're impossible to answer almost immediately, some of them, although one of the ones [in the film] is easy for an American audience to answer, but impossible for an Indian kid to answer. There's a question about the American $100 bill. So all I can say about the story is that if it isn't true it should be.
The interviewer, later, wrote a column that gave a bit more detail:
Screenwriter Simon Beaufoy ("The Full Monty") took considerable liberties with the original novel, which is based on an apocryphal story about a teen from the slums of Mumbai who wins 20 million rupees (roughly $400,000) on "Who Wants to Be a Millionaire."
Trying to find out the truth of the tale in India was near impossible, Boyle found. "There's a billion people racing ahead in a 10% growth economy," he says, "enough to start up a decent-size planet from scratch. Did a slum kid go on the show? There are lots of stories. Educated people went on pretending to be rickshaw drivers. If the story isn't true, it should be."