1: I could have not answered this question, but...
Both OP's example and mine are completely grammatical, but it's worth pointing out that even native speakers (especially when not paying close attention) could be liable to misparse mine as...
2: I could not have answered
3: I couldn't have answered
...because "could not have" is far more common than "could have not".
In my specific example there's a clear semantic distinction between my first version (It would have been possible for me not to answer) and the more common forms meaning It would not have been possible for me to answer.
That same ambiguity potentially exists in the present tense form...
4: I can not answer
But in practice that version would carry the strong implication not answering is an option available to me, because if the intended sense was I am not able to answer it would probably be written as...
5: I cannot answer
6: I can't answer
In speech, the less common sense could be conveyed by stressing either can or not (if only one of the two words is stressed, it steers the audience away from the one-word cannot version).
Another common way to steer the audience away from an unwanted misparsing is...
7: I could simply not have answered
8: I could have just not answered
...where introducing another word between could and not again steers the reader/audience towards the intended sense.