Although might may have become the main exponent of epistemic possibility in every day spoken English, it is generally not synonymous with may, as the following example, borrowed from Charreyre (1984: 55-56) illustrates:
in Just look at the sky! There might be a storm soon, uttered in front of a threatening sky, it would be contextually less appropriate, Charreyre suggests, to use may instead of might, while will would be perfectly acceptable. May would convey very little information and simply express the logical, theoretical and objective possibility that such a sky is likely to bring about a storm, which is obvious. Might, on the other hand, carries some appreciation on the part of the speaker that, although a storm may not be wished for, yet it must not be ruled out.
-- Towards a contextual micro-analysis of the non-equivalence of might and could by Stephane Gresset
Claude Charreyre is a French linguist, but I'm not sure his assessment on might vs. may can represent how a native speaker of English would think about it.
Do you think his conclusion was stretching it? Why?