I meant when people say "Jamie and I really enjoyed that today". It should be Jamie and me. But people are trying to talk "posh". The old joke, a primary school teacher asked a pupil to say a sentence using the word "I". The pupil answered "I is..." The teacher interrupted saying "always use 'I am', not 'I is' ". The pupil replied "I am the letter after H in the alphabet".
But if you took Jamie out of the picture, you wouldn't say "Me really enjoyed that today." Unless, perhaps you were Tarzan. 'Jamie and I' is correct in your example.
It will be 'me' when I am the object of the sentence and 'I' when I am the subject. So:
Jamie and I were invited to Fred's party
. That's correct, as 'Jamie and I' is the subject of the sentence.Fred invited Jamie and me to his party.
Here 'Jamie and me' is the object, and Fred s the subject.
The trick for working it out is to remove Jamie from the sentence, and whatever would be right for you alone, will be the right one to use when you're being grouped with others.