She may/might be honest.
This sentence already has the meaning that
She may/might not be honest.
1. Why do we sometimes say:
She might and/or might not be honest.
2. Which conjunction is (more) correct: "and" or "or"?
So, the question is not on when to use "might" and when to use "may", it is about why each comes along with its negation, and what difference that usage makes to the meaning. Is it just about emphasis?
The second question is about the conjunction that should be used between them: "and" or "or"? Both versions are used frequently on the web.