Unless it is an acoustic guitar string (nylon or wound with non-magnetic material) or clearly not vibrating, it's the pickup that's broken: any wiring possibly connected to the strings is just to reduce hum, not change the signal level. The pickup reacts perfectly fine to an entirely unconnected string if it is an electric guitar string rather than an acoustic one.
I think that the coils are usually wired in series, so if you still get sound from the other strings, there are rather few error scenarios I can think of:
a) coil is shortcircuited. This will most likely be purely mechanical: two items touching that shouldn't.
b) magnet is dead and/or has fallen out or there is some other break in the magnetic connection in the pickup
c) someone disassembled the pickup at some point of time, the pickup is a double-coil pickup, and one of the coils or one of the magnets was put back in reverse. Double coil pickups are intended to pick up outer magnetic fields in a manner that the voltage inducted into both coils cancels while the voltage induced from field changes done by the vibrating string to the field from the coil magnets adds up. Reverse a coil, and you pick up the noise and cancel the sound. Reverse a magnet, and both noise and sound cancel.