I would disconnect the neck pickup from the guitar completely and measure the resistance through it with a multimeter. If it's a 4-wire pickup, check to make sure that the two wires that are supposed to be soldered together & taped (in a stock installation) are making a good connection. If it checks out, then re-install it and use your multimeter to check the circuit continuity on both sides of the pickup--the signal side all the way to the tip of the output jack, and the ground all the way to the sleeve of the output jack. Keep in mind that a good ground connection is just as important as the "hot" connection in getting your signal through a pickup.

A couple of good close-up photos would be helpful in visualizing what's going on. They must be well-lit and well-focused. And BIG.