It really depends on what you're getting at!
There are an arbitrary number of methods for determining whether a password is similar to another one. Let's say for example that you use a password card, and that somehow someone else has the same one (or simply knows which one you have). If they compromise one of your passwords and can see that it's just a row down the password card, they're likely to guess (maybe even correctly) that your passwords are all derived from that card in a similar fashion.
But, for most things this really isn't an issue at all. If your password on service A differs from the password on service B by only a single character, and both services are secure (e.g., store salted hashes for your password instead of the straight hash or the plaintext itself) then it is “computationally infeasible” to determine whether the passwords are similar, let alone how similar they are.
A short answer is this: If your passwords follow any sort of pattern, then yes, it is likely that the compromise of one password will lead to the compromise of others. However, that doesn't mean that it's going to be feasible to do so. As long as you:
- Never use the same password for more than one service,
- Introduce some random (even if only slightly) element into the generation of your passwords, and
- Never transmit or save your passwords in cleartext
You should be just fine. And remember to always have different passwords for different services—don't simply use the same password for everything, and do not even use the same password twice. It is important to guard against stupid companies that refuse to follow best practices when it comes to the storage of user data such as passwords.
