Originally Posted by Loud Noises
... If I turn the tone knob to 0 the hum is gone because the pickups are wired to be hum canceling.
That's not correct. The hum goes away when you turn the tone down because the EMI (electro magnetic interference) is being shorted to ground. There's nothing wrong with your tone control or tone pot. J-basses are only hum cancelling when BOTH PICKUPS ARE ON. Unfortunately, what you're experiencing is single-coil hum and it can't be completely eliminated. Shielding can help, so can proper grounding of your electrical system (it should be properly grounded anyway). Neon signs, flourescent lights, CRT monitors and so forth are bad sources of EMI and sometimes they are just unavoidable at a gig.
The best solution for me is any type of double-coil humbucking or hum-cancelling pickup, they sound fine to me but some people prefer the "true single coil" tone.