When all the adults are arguing in the side room after Harry's name came out of the Goblet of Fire, Moody suggests that someone confunded the Goblet to think there were four schools, and submitted Harry's name as the only candidate for that school:
“Because they hoodwinked a very powerful magical object!” said Moody. “It would have needed an exceptionally strong Confundus Charm to bamboozle that goblet into forgetting that only three schools compete in the tournament. ... I’m guessing they submitted Potter’s name under a fourth school, to make sure he was the only one in his category....”
That this is in fact what happened is confirmed when Moody tells this to Harry in his office after Harry returns from the graveyard:
“Who put your name in the Goblet of Fire, under the name of a different school? I did.
Throughout the rest of the book Harry is considered (one of) the Hogwarts champion(s) and is not considered the champion of the unnamed fourth school.
This would seem to indicate that a student is the champion of the school he attends, regardless of the school that he entered under.
Theoretically, then, students could submit their names under different schools and the Goblet will pick one student with each school name, and we could end up with all three students from the same school.
For example, three students from Hogwarts enter the tournament. One writes "Hogwarts", one writes "Beauxbatons", and one writes "Durmstrang". The Goblet decides that the best candidates from Beauxbatons and Durmstrang are actually the Hogwarts students and they get picked. Since all three students are from Hogwarts, Durmstrang and Beauxbatons now have no champions, Hogwarts automatically wins, and they can all just go home.
Could something like this actually happen? If not, why not?
If yes, this seems like a really easy way to have a rigged tournament.