When server you want to connect to receives more than MaxAuthTries (default = 6), it falls back to regular password authentication. To prevent this, you can specify which key to use for what server in two ways.
As cmd option:
ssh -i ~/.ssh/id_rsa user@hostname
By specifying Host/IdentityFile pair in ~/.ssh/config:
Host hostname IdentityFile /home/USER/.ssh/id_rsa Host hostname2 IdentityFile /home/USER/.ssh/id_rsa2
Of course you can also increase MaxAuthTries value on the SSH server(s) in /etc/ssh/sshd_config, but this is not recommended!
This automatic mechanism is also annoying when using password authentication. To force the ssh client to use password only and not to use keys, use PreferredAuthentications option:
ssh -o PreferredAuthentications=password hostname
(This post is just an extended answer from http://serverfault.com/questions/36291/how-to-recover-from-too-many-authentication-failures-for-user-root/256083#256083 )