| When you loosen the screw (screw rises, pickup stays put) try lifting the pickup with your fingers to make sure nothing is holding it. If it comes up easily, then whatever "spring" type it has under it isn't doing its job. If it doesn't come up easily, the pickup wire might be binding in the wire's hole. You might have to push some wire slack up through the hole to fix this.
Most times, there is either "foam" pads on the bottom of the p'up or "springs" that push the p'up UP when you loosen the screws. With the foam pads, sometimes they've been compressed down so long that they don't spring back and push the p'up up.
It's usually an easy fix, by removing the pup, and adding a shim (foam, cardboard, etc..) under it. This will help push the pup up when you loosen the screws.
My bass had this problem. I cut cardboard pieces to fit the pickup slot and this added just enough to keep the pickup up good and snug, while still allowing me to lower it without stripping the screw.
First, measure the pup's height from the bass body so you can get it back close, then remove the strings, then remove the pickup to find out what's holding it or if new springs or a shim is needed.
Shouldn't be too hard to fix.
Mag... | 
