Yesterday, I asked one of my students how he was. He answered, "I am more better." I told him that "I am (much) better" would do. He answered, I was already better yesterday, and I am better than I was yesterday." It made me think twice. Was I wrong with my correction?
Learn English – Better, more better or much better, which is correct
comparative-degree
Related Solutions
SHORT VERSION:
Don’t use it.
LONG VERSION:
In your first example not quite better is not parsed as [not][quite better] but as [not quite][better]; that is, not quite is a fixed expression meaning ‘almost’, ‘falling just short of’. This is fine.
But this not quite does not behave the same way as bare quite. Quite ADJer is and not unknown, but it is very rare, comparable to very ADJer. Take a look at this Google Ngram, where I have multiplied the instances of quite better and very better by 1000 and they are still outranked by much better:
I note that your second and third examples are uttered by a native speaker of Italian and a speaker of Indian English.
There is as far as I know no reason why very and quite should not be used with comparatives; but they aren’t.
This is not a rare use: it implies that Havel, although not perhaps one of the most fascinating 20th-century politicians, was among those who were more fascinating than ordinary politicians.
Best Answer
No, you aren't wrong.
The simple rule is this that you use 'much' ( instead of very, more etc ) with 2nd degree of adjective.
What your student should have said was this,
or,