Weak how? Some Fenders (and other makers) have dead spots, with Fender having a "knack" for one on the 6th fret, but I've always noticed it on the g string.
I'm not sure how a Precision pickup could have a weak area, especially on one string (I think the G string would be affected as well, but I'm no expert). Either the pole pieces aren't aligned with the strings or it's not setup properly or the pickup need to be raised to compensate for the neck radius.
Just my $0.02