It's quite simple really. Consider these examples:
It might be a lie.
It could be a lie.
In these two statements, the first makes room for a 'but' in the equation, such as, 'It might be a lie, but it helped us succeed'. Here the first statement is an admission of the lie. It explicitly states that what you said might be a lie, but it was still beneficial in some sense, so it's acceptable.
Whereas the second statement, make no such admission of guilt. It just states that it could be lie, it couldn't be a lie and such.
I feel this 'could vs might' dilemma only happens when you're using them in a negative sense like, 'It might be a lie', 'He might have killed him' and such.
Anyway, this is just my understanding of the concept, please feel free to debate or message me.