I recently had this problem with a Broadcaster pickup sent to me by long time friend Sid McGinnis who has played guitar on the David Letterman show since 1984. As I examined the pickup there appeared to be no physical evidence of problems from the outside. As I measured the pickup I noticed the DC resistance would change upon pressing on the low E magnet. I then chilled the pickup and noticed the resistance would go lower then quit. As it warmed up the pickup would have a high DC resistance and then drop and stay constant around 4.7 to 5.2 K ohms DC. This was too low for an early Broadcaster pickup. So I decided to remove the plain enamel wire turn for turn till I got to the problem.
As I nearly removed all the wire, I began to see the problem within the bobbin. The end magnet had a crack in it from some unknown reason. This can happen when using a dull chisel to remove an old coil and the excessive side pressure can crack the end magnets. Sid's bobbin was original and I had to fix the problem. We carefully removed the end magnet from the bottom flatwork as it was originally pressed into place with the chamfer on the magnet facing the top of the bobbin.
I found another old Broadcaster magnet within my boxes of parts and demagnetized it because the chamfer side had an opposite polarity. Some old pickups were North polarity and others of the same time period were magnetized South. At times the phasing didn't matter with the old pickups and depending on how the switch was wired, or if it had a blend control. As I replaced the broken magnet and hand wound the coil with the original 50 year old plain enamel magnet wire. I had to remagnetize the single pole without changing the gauss on the existing ones. I made a special fixture with pointers to touch the top and bottom side of the unmagnetized rod magnet.
I carefully placed two custom made coils around each end of the pointer and charged the rods with DC current and magnetized the pole piece to the proper polarity. It was a little too strong, so I applied a controlled amount of AC current to the same coils to degauss the pole. The original cracked magnet would bite the magnet wire and depending on the temperature the contraction and expansion made the pickup work and then not work. It was a tricky job but well worth saving a great pickup for a great guitar.