If you accidentally take your neighbors mail out of the mailbox, is it legal to put it back in?

Yes, and you should do so as soon as possible.

Was it mis-delivered and placed in your mailbox by mistake? It so, not only would it be courteous, but legal to put it in your neighbor's box and you should do immediately. If it wasn't in your mailbox, the questions arises as to why you were in your neighbor's mailbox. Remember: If it isn't addressed to you, don't open it. It is against Federal Law to open someone else's mail.

There is nothing illegal about putting mail in their box.