I have seen this happen quite often. Care must be taken when the covers are removed as the solder that is on the cover can snag the tape and coil wires. Most often it can be the taped connections between the two bobbins or connections to the shielded lead wire that's connected to the bottom plate. If the connections become exposed, they can touch the cover or bottom plate and short out. Make sure all the connections are insulated.
I have seen this happen a lot when putting covers on humbuckers with four conductor wire. It can get pretty tight when trying to slip a cover over the pickup with all the tape and wires. I have seen a drop in output as one coil can be shorted out. Just be careful or contact an experienced repair man. I have done silly things too. I have checked pickups with an ohm meter and the pickup showed a dead short. I took the pickup apart and it worked great. When I reassembled it, there was that dead short again! What I forgot to check was the shielded cable to the controls. I found that a thin strand of shielded braid (ground wire) was touching the hot conductor and caused a complete short.
I have checked single coil pickups with the same mistakes. I checked the pickup with an ohm meter and again there was a dead short! I found there was excess solder on the hot connections that were shorted to ground. After a while of making mistakes, you learn to check everything.