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Maybe update your sshd if it is an old version or downgrade if it only happens with a new version etc.
You could also compare /etc/ssh/sshd_config of this and a working server as a start.

Edit:...

Is this the case no matter from where you login? Do you have another box you could try to login from? If the problem persists, maybe restart the sshd. If this also does not help you could issue your...

ssh-copy-id