So if you have a similar problem with your "generic" bass or guitar, you may find this instructable useful.
Of course I had to make sure that the problem was not with other components of my setup, so I replaced cables, tested the bass with my son's guitar amp (at low volumes -- I didn't want to buy him a new one) and even used a battery-powered headphone amp, but the radio station was still there. I eventually plugged my son's guitars to the bass amp -- lots of hum, but no RF interference. So please test your setup and make sure the problem is with your instrument before trying anything, otherwise you will only frustrate yourself!
Now I'll show you the method used to solve the problem, which was surprisingly effective considering the simple and inexpensive materials I used and that it took me slightly over 3 hours. So let's get started!
Important: be sure you are handy with tools and have good soldering skills. Don't blame me if anything goes wrong!