Sshd refusing all authentication methods

I have a server which I suddenly can't connect to with ssh. sshd is obviously running and responding, but won't take anything. It doesn't even ask for a password, it refuses the method itself.

Other services on it still appear to be running. It still reports its status. But it's not serving network customers.

What would cause sshd to do something like this and still run? What might have happened?

On further consideration I think a disk may have filled up. Unfortunate.

Yup, I faced the same issue some time back and my disk was full. The problem was solved after doing some disk cleanup.