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From an online chat interview (found here), J.K. Rowling answers a question:
Q. In the first book you said Slytherin house Quidditch captain was sixth year Marcus Flint. If there are only seven years of Hogwarts, why is he in the third book?
A. He had to do a year again! :-)
Indicating that he failed his N.E.W.T.S. and had to repeat the year.
Apart from Marcus Flint is there any other mention of a student being held back a year for failing their classes?
As far as I'm aware, no-one else is actually forced to retake an entire year. Failing a year is threatened as a consequence for failing exams. Even in year one, the threat of being held back is there.
"Anyway, what are you revising for? You already know it all."
"What am I revising for? Are you mad? You realise we need to pass these exams to get into second year? They're very important, I should have started studying a month ago, I don't know what's got into me..."
(Philosopher's Stone, Chapter 14, Norbert the Norwegian Ridgeback).
However, the standard of the exams seems to be sufficiently low that even the real dunces are able to scrape through.
Harry had almost forgotten that the exam results were still to come, but come they did. To their great surprise, both he and Ron passed with good marks; Hermione, of course, came top of the year. Even Neville scraped through, his good Herbology Mark making up for his abysmal Potions one. They had hoped that Goyle, who was almost as stupid as he was mean, might be thrown out, but he had passed, too. It was a shame, but as Ron said, you couldn't have everything in life.
(Philosopher's Stone, Chapter 17, The Man With Two Faces).
This would suggest that Flint really was on another level of poor performance since even Crabbe and Goyle never missed a year.
The passage also hints that really terrible students may be forced to leave Hogwarts altogether. However, we never see this happen so this may just be speculation on the trio's part.
Crabbe and Goyle aren't held back a year but, as amflare says, they do have to resit their DADA OWLs.
"If your friends Crabbe and Goyle intend to pass their Defence Against the Dark Arts O.W.L. this time around, they will need to work a little harder than they are doing at pres-"
(Half-Blood Prince, Chapter 15, The Unbreakable Vow).
This shows that, in later years, students can resit individual subjects if they choose without necessarily having to redo the entire year. As far as I can tell, even Crabbe and Goyle avoided that fate.
As far as I know this is not confirmed anywhere, but a possible student that got held back would be Sally-Ann Perks. Consider the following:
In Harry Potter and the Philosopher's Stone she was sorted right before Harry.
There weren't many people left now.
"Moon" ... "Nott" ... "Parkinson" ... then a pair of twin girls, "Patil" and "Patil" ... then "Perks, Sally-Anne" ... and then, at last —
"Potter, Harry!"
In Harry Potter and the Order of the Phoenix students are called into their Charms O.W.L in alphabetical order, yet Sally-Anne Perks is mysteriously absent when the Patils and Harry are called.
Ten minutes later, Professor Flitwick called, "Parkinson, Pansy — Patil, Padma — Patil, Parvati — Potter, Harry."
This tells us that she was not present at the Charms O.W.L. As Charms is a mandatory course, she can't have not taken the class. While it is possible that she missed the exam for some other reason (e.g. she was in the Hospital Wing with some dreadful injury), perhaps she was not there because she was held back.
(This could also explain why we never hear of her again after the sorting – if she was held back, Harry and his friends would have much less to do with her.)
I doubt that Marcus Flint was held back a year, J.K.Rowlings reply was an off the cuff remark to give an answer without having to explain the inner working of the British Education system. But Harry Potter broadly mimics the way that public schools work here.
Sixth Form in British Education
The final 2-3 years of school education from 16-18 are described as the Sixth Form or the Sixth year, with the first of the 2 being the Lower Sixth and the second the Upper Sixth. A student in either the Upper or Lower Sixth would be described as a Six Former or a Sixth Year.
