harry potter - Have any students not been sorted?

Based on an answer on Ilvermorny school's sorting ceremony, multiple houses of the school can express an interest in a student, and some students have received a request for all four houses.

Have any students ever not been chosen by an Ilvermorny house, or any wizarding schools? Have there been any Hatstalls to the point of rejection once arriving at a school?

Answer

Neither at Hogwarts or Ilvermorny is there any mention of a student who is actually supposed to be attending the school but isn’t chosen for any house. Harry does worry that he wouldn’t be chosen for any house, but there’s nothing implying that it’s actually a possibility.

“A horrible thought struck Harry, as horrible thoughts always do when you’re very nervous. What if he wasn’t chosen at all? What if he just sat there with the hat over his eyes for ages, until Professor McGonagall jerked it off his head and said there had obviously been a mistake and he’d better get back on the train?”
_{^{- Harry Potter and the Philosopher's Stone, Chapter 7 (The Sorting Hat)}}

This seems likely to be Harry’s own worries rather than something that actually would happen at Hogwarts, and he didn’t know about Ilvermorny so it wouldn’t be applicable there.

In Hogwarts, it seems likely that any student who doesn’t really fit into any of the houses would go to Hufflepuff, as Helga Hufflepuff was willing to take all students into her house.

“Said Hufflepuff, ‘I’ll teach the lot,
And treat them just the same.”
_{^{- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)}}

The Sorting Hat was intended to continue choosing the students for each house in the same way the Founders did, and Helga Hufflepuff was willing to take the students not chosen by any of the others. Presumably, then, students who don’t fit anywhere would still go to Hufflepuff.

“Good Hufflepuff, she took the rest,
And taught them all she knew,

Thus the houses and their founders
Retained friendships firm and true.”*
_{^{- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)}}

Because of this, it’s highly unlikely that any students at Hogwarts wouldn’t be Sorted into a house, as if they don’t have any of the four houses’ traits, they’d likely be just put into Hufflepuff.

It’s a bit more difficult to deduce exactly what would happen with a student in Ilvermorny who’s ill-suited to all four houses. There’s no house in Ilvermorny that clearly would be willing to take the students who don’t really fit in anywhere. There’s nothing that indicates that there’s any case where no carving lights up to take a student, however, so there may be something that ensures at least one of them will ‘choose’ them even if they’re not a good fit just so all students have a house. In Ilvermorny, they also have inter-house competitions, so any students not being placed in a house would complicate matters as they’d have no house to compete for.

The house expanded with every passing year. More students had arrived and while the school was still small, there were enough children to fulfil Webster’s dream of inter-house competitions.
_{^{- Ilvermorny School of Witchcraft and Wizardry (Pottermore)}}

Though it’s unclear exactly what it is, there’s presumably some way that Ilvermorny ensures all of its students are accepted into one of the houses, even if no house really ‘fits’ them. As Valorum suggests in the comments, it’s possible that the student then gets to choose, like they would if they were chosen to be in more than one of the Ilvermorny houses.

The schools other than Hogwarts and Ilvermorny aren’t said to have houses. Since the other known wizarding schools don’t seem to have houses, there’d be no chance that a student wouldn’t fit into a house - all students would simply attend the school and none of them would have houses.