I’ve seen this quite often and recently worked on Neil Young’s pickup with the same problem. The problem most likely comes from the older shielded hookup cable that is soldered to the controls and ground from the pickup. The inside multi-strand conductor is coated with a rubber based insulation to keep it insulated from the outer braided shield. The inside rubber insulation begins to crumble and break down with age. As the cable is moved the insulation breaks and can cause the inner multi-strand conductor to short against the outer shield. While playing if there is any movement it can cause the two wires to touch causing the pickup to cut in and out. You will need to have the cable replaced with a cloth braid shielded cable as used on vintage style humbuckers. This normally doesn’t cause any tone change and will keep the pickup working for many years. I would suggest checking all the other hookup wire for the same problem.