Well, a major theme in the Harry Potter series is the importance of choice and free-will. That's what the last two to three pages of the Horcruxes chapter in HBP was about.
The Sorting Hat is an object that delves into a person's mind, judging them on what it understands about the witch or wizard's personality, skills, abilities and other qualities. However, the moral lesson here is that, as Dumbledore says, it is our choices that show what we truly are, far more than our abilities. So the Sorting Hat can oblige a student's choice of House, as it did for Harry.
Hermione Granger states in OotP that the Hat did seriously consider placing her in Ravenclaw (probably because of her intellect), but had to take a moment before deciding on Gryffindor. Perhaps it realised that she was more brave than brainy? Or something along those lines. Dumbledore, during a flash-back in DH, muses to Snape that "sometimes I think we Sort too soon," in reference to Snape's bravery at refusing to flee Hogwarts like Karkaroff. Perhaps the Sorting Hat thought of placing Snape in Gryffindor during his Sorting but placed him in the House he wanted to be in: Slytherin? That's speculation, but I think, legitimate.
As to your question, though, we never really get an example in the books of a student who explicitly asks the Hat to be placed in a House of their choosing and then fails to get their wish. I think it's fair to assume that it is possible, however. Generally, though, the Hat seems a nice-enough character to respect peoples' choices.