I think you're right on the money, mate.
First up, I'd look at the most simple things: look for stray bits of wire or solder that might be causing part of your input signal from that pickup to short to earth. Look for dry or messy solder joints in that signal path that might be causing voltage drops.
After checking all your connections, try soldering the suspect pickup straight to the output jack. If it works, then you can rule it out as the problem.
If you have a multimeter, then it's really easy to check that your switches and pots are working properly by checking their resistance and continuity as you flick the switch or turn the knob. (You'll probably have to take them out of circuit to do this, though).
So basically, I'd check the simple things, then try to eliminate the pickup itself as faulty, then rewire it in and redo all the connections, checking each component with a multimeter as you go. You might find that simply redoing solder joints will fix your problem.
Good luck!
