Can/Could vs May/Might

There are two categories of modality: deontic and epistemic. For example:

Can/may I go to the bathroom? = deontic (permission) It can't/may not be true! = epistemic (deduction)

Could/might I ask a question? = deontic (permission) Could/might it be true? = epistemic (deduction)

So maybe this influences whether the sentences sound 'weird'! For example 'Might I ask a question?' sounds formal to me, but 'Might it be true?' doesn't. 'May' also sounds more formal than 'can' with deontic use, and more tentative for epistemic use.

For more information see Michael Lewis' 'The English Verb'. He calls 'could' and 'might' the 'second forms' of 'can' and 'may', which means that they are used not only to refer to past time but also to ideas distanced from reality (e.g. conditionals) and distanced psychologically (e.g. polite requests). Thus 'Could I ask a question' is more polite than 'Can I ask a question'.

I hope that helps!