1. He could have given me some advice.
(Does this sentence mean that he didn't give me some advice?)
2. He cold have told a lie.
(Does this sentence mean that he didn't told a lie? Or does #2 mean #3?)
3. He may have told a lie.
1. Right, he didn't give advice even though he could have.
2. I'd take this to mean he could have but didn't.
3. This is different. There is doubt whether he told a lie or not.