I would say that if it's a general rule that extra characters are always ignored, then yes, there's a security problem in there somewhere. Namely the fact that the system would either have to store your password length (so it knows when to start ignoring) or not use a hash, both of which are way less secure if/when the database gets breached.
If, however, it's as @gowenfawr suggested, that there's simply a maximum length of password, so any passwords longer get truncated, then that's only a little bit less secure (and no less secure than simply rejecting passwords longer than the maximum).
To explain the last parenthetical statement, let's say your system accepts passwords up to only 10 characters long (every system has a maximum, though I'm not sure why... that's another tangent). The usual method is simply to reject any password a user tries to create that is longer than that. Thus, this maximum is a known variable (a serious hacker may likely first create an account to find out the password limitations), and hackers need only to try passwords that long or shorter.
On the other hand, if the system accepts passwords of any length, but quietly truncates them, it would be slightly harder (but still not impossible) to figure out that that was going on. So if you consider more obscurity to be more secure, then the truncation method is actually more secure than the rejection method... unless both are being attacked by someone who is unaware of the maximum... then the truncation method would be slightly less secure if the attacker starts with longer passwords.
As @nobody pointed out, the situation where someone is relying on the length of their password to provide their security (as in pass phrases), them not knowing it's being silently truncated greatly reduces the security, since the password they thought was secure, "Passwords often 16 variable wrench sanctification for 53 sets behind morbid technicalities!" ends up getting truncated to a very insecure one, "Passwords "