| It's quite normal for mains buzz to appear to go away when you turn down the tone control. The tone control on a passive bass just bleeds treble to ground. The more you turn it anti-clockwise, the more treble gets dumped - so the high-frequency bits of the interference (which is what your ears notice) get quieter.
As for the cause of the problem, you've pretty much worked it out for yourself. Somewhere in your house there is a major source of RFI (interference), and your bass is picking it up. There's nothing wrong with your bass - although shielding the cavities would help reduce RFI quite a bit.
Possible sources of noise:
Do you have any lighting dimmers in the same room you play in?
TV screens / CRT monitors?
Fluorescent tubes?
One thing for sure: it won't be anything to do with your wireless router. Halogen lamps may be a possibility, though.
If your bass cord is long enough, walk around the room with the bass plugged in and the tone control up. Point the bass at various electrical devices and listen. One (or more) of them will soon show up as the main culprit. Once you've found it, switch the damn thing off! | 
