Several times, in the Harry Potter series, we are led to believe the sortee has some control over the House they are sorted into. Most famously, Harry Potter turns down his chance at being in Slytherin House, as suggested by the Sorting Hat to be in Gryffindor.
Harry: Not Slytherin. Not Slytherin.
Sorting Hat: Not Slytherin, eh? Are you sure? You could be great, you know. It's all here in your head. And Slytherin will help you on the way to greatness, there's no doubt about that. No?
Harry: Please, please. Anything but Slytherin, anything but Slytherin.
Sorting Hat: Well if you're sure, better be... GRYFFINDOR!
Retrospectively, he fits in Gryffindor House better, but it begs the question:
How far can you persuade the hat away from the House it wants to sort you into? The Hat is quite adamant that Harry would do well in Slytherin. Could a student convince it to sort into the 'wrong' house entirely?