How strong is strong? The best way to find out is to launch a dictionary attack on your own system to find out! john is a package (and a command of the same name) that can help you assess password strength.
Note that john still needs access to the /etc/shadow password file to work. This means that a normal end-user cannot use john to crack the passwords of other end users.
You can copy the /etc/shadow file from a slow system to a faster system to speed up password cracking.