First of all, I am not an ethesist, even though I may play one on the internet (see, I can't even spell it).

Secondly, I would think that when one notices a mistake, they do have a moral responsibility to make the other aware of it. Same as getting too much change back at Burger King. Why don't you just send them an email that says "I just checked my frequent flyer account and I think you might have deposited too many miles into it It is account 1234abcd."

At that point, I think that you have done the morally correct thing. If they want to persue it, they can. I do not think that you have any obligation to make sure that they deduct the miles. If they are still there after a statement cycle or two, I would think that you can keep them.