FIXED
If you have seen this in the login screen of your server...
stdin: is not a tty
...you know the trouble. Well. I've had the exact same problem on my root server. I tried several things to remount /dev/pts
or /dev/tty
but nothing worked and I wasn't able to open a second connection.
Then a friend of mine had the most basic idea to just restart the service "screen", which allows to create multiple virtual console sessions.
Guess what. The command...
service screen restart
...fixed the console sessions to work again. At the time the error occured, we had appox. 20 screens running.