Can a Hogwarts student refuse the Sorting Hat's decision?

Question

After reading Harry Potter and the Cursed Child,

and seeing how Harry's son was assigned to Slytherin, even though he did not want to be,

and remembering how Harry asked the Sorting Hat not to join Slytherin and how it granted his request, I started wondering what happens when a student refuses to join a specific house.

I am not asking about the process when students can say what they want or don't want, and the Sorting Hat might take that into account, but about when the choice has already been made.

Obviously, not everyone gets what they want. But if, for example, Draco was sorted into Hufflepuff (my head hurts from imagining him in yellow robes) then I am sure he would not take that so lightly (and, of course, his father would hear about it). Could he, or anyone else, refuse to be a part of a house they do not want to be in?

What would the consequences be?

Answer 1

No, it doesn't look like it.

From Pottermore, we see that the Sorting Hat is very stubborn about sticking to the decisions it arrives at. It seems that it's impossible to protest that the Sorting Hat got the decision wrong and to ask it to reconsider.

The Sorting Hat is notorious for refusing to admit it has made a mistake in its sorting of a student. On those occasions when Slytherins behave altruistically or selflessly, when Ravenclaws flunk all their exams, when Hufflepuffs prove lazy yet academically gifted and when Gryffindors exhibit cowardice, the Hat steadfastly backs its original decision. On balance, however, the Hat has made remarkably few errors of judgement over the many centuries it has been at work.
_{("The Sorting Hat", Pottermore).}

Of course, the hat does sometimes take a very long time to arrive at a decision. As the question says, it frequently takes into consideration what the student themselves would like. Where the person and the hat remain in conflict, it seems likely that the hat makes the final decision, as was the case with Neville.

Of Harry Potter’s contemporaries, Hermione Granger and Neville Longbottom came closest to being Hatstalls. The Sorting Hat spent nearly four minutes trying to decide whether it should place Hermione in Ravenclaw or Gryffindor. In Neville’s case, the Hat was determined to place him in Gryffindor: Neville, intimidated by that house’s reputation for bravery, requested a placing in Hufflepuff. Their silent wrangling resulted in triumph for the Hat.
_{("Hatstall", Pottermore).}

Whether the hat effectively chose on Neville's behalf or whether, after the hat dug its heels in, Neville finally relented and gave his consent to be put in Gryffindor is unclear. At that stage, Neville isn't exactly a strong-willed person who'd dig in his heels in a confrontation (when he gets older it'd maybe be a different story) so it may be that we don't have a clear example of someone who's truly prepared to stand up to the hat. Nevertheless, once the hat has shouted out the name of the house there doesn't appear to be anything the student can do about it. You have to go to your allocated house and make your peace with where you've been put. There isn't really much point in having a magical hat to decide who goes in which house if the decision is open to negotiation.

Even when it's a very tight toss-up between two different houses, the Sorting Hat sticks to its guns. It was still insisting that Pettigrew belonged in Gryffindor years after its original decision.

The only true Hatstalls known personally to Harry Potter were Minerva McGonagall and Peter Pettigrew. The former caused the hat to agonise for five and a half minutes as to whether Minerva ought to go to Ravenclaw or Gryffindor; the latter was placed in Gryffindor after a long deliberation between that house and Slytherin. The Sorting Hat, which is infamously stubborn, still refuses to accept that its decision in the case of the latter may have been erroneous, citing the manner in which Pettigrew died as (dubious) evidence.
_{("Hatstall", Pottermore).}

So the hat's decision is final and there doesn't appear to be any way to challenge it.

Answer 2

There is actually an example where a student theorized about what he would do if placed in a house he didn't want to be in. From Chapter Nine of Chamber of Secrets:

"I always knew Salazar Slytherin was a twisted old loony," Ron told Harry and Hermione as they fought their way through the teeming corridors at the end of the lesson to drop off their bags before dinner. "But I never knew he started all this pure-blood stuff. I wouldn't be in his house if you paid me. Honestly, if the Sorting Hat had tried to put me in Slytherin, I'd've got the train straight back home....

If we assume that Ron is correct, it seems like there would be no recourse other than to leave Hogwarts if you don't like your house. If it was possible to appeal the Hat's decision, or switch houses some other way, it wouldn't really be necessary to take the train straight back home.

Similarly, in Chapter Five of Philosopher's Stone when Draco Malfoy discusses getting sorted into a bad house he also only mentions the option of leaving:

"Well, no one really knows until they get there, do they, but I know I'll be> in Slytherin, all our family have been – imagine being in Hufflepuff, I think I'd leave, wouldn't you?"

Once again it seems that there may be nothing else you can do once the Sorting Hat issues its proclamation (though I suppose it's possible that Malfoy would leave not because there's nothing to do about it but because his honor was slighted).