Would re-soldering the wires fix this problem?
unfortunatly i don't have a multimeter
I had this problem when I installed my current set of pickups into the LP. I accidentally melted part of the plastic wire casing off the wire running from the pickup selector switch to the volume pot, enough that the wire could touch the volume pot and cause it to have almost no volume sometimes, essentially grounding the hot wire of the pickup. Shaking the guitar would fix it for a little while. There is not much extra slack on the wire so it won't stay too far away from the pot. A little bit of black electrical tape solved this...
How I identified the problem was to take the cover off the back compartment and I couldn't tell if I actually melted enough of the wire sleeve by looking at it, but when I pulled the wire away from the pot, it worked fine so I knew what the problem was.