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Agree with Bhai that this is not the GMAT type. The answer is E (or A - both are same....there are only 4 choices for this question.). Anyways, I think the question is to especially elaborate the foll concepts.
1. Look at the second part of the sentence. Technically you can rewrite it as 'I was the thief'. You cant rewrite it as 'me was the thief'. So, 'I' is right for the second part.
2. the first part of the sentence is tricky...'but' is a preposition here and not a conjunction. Why? becuase, you can rewrite the first part as 'No one could have told them that, but him'. This makes 'him' right too.
I think that the answer is A . This is an example of pronoun-case error. In this sentence 'but' plays the role of a preposition. so any pronoun following a preposition must be in object case and hence the him. Also, any pronoun following 'is' or 'was' must be in subject case and hence the I.