What impact does it have to accept another answer after answering one already?
Can I accept another answer without "refusing" the first one?
Is this acceptable etiquette?
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There is nothing wrong about switching up who you select as an accepted answer. It is perfectly logical to assume that a better answer can come along. I would suggest, however, not pulling the trigger on an accepted answer unless the answer actually answers your question to satisfaction. It can make the question asker look a bit indecisive if they keep changing the check mark around.
If you accept a different answer, the originally accepted answer will no longer be accepted. The user of the original answer will lose the 15 points they gained because of your acceptance and the answerer for the new accepted answer will gain 15. Hopefully what this means about the person "reaccepting" is that they came back to a question they asked and found an even better answer.
*Note the points would not apply in a CW (Community Wiki) situation
In addition to what others have said, you can always up-vote as many answers as you like, so if you get two answers that are good, helpful, and well written, but one is a little better than the other, I'd say you should up-vote both of them, and accept the better one, even if it came along later, and you had already accepted the first one.