I plug my bass in and it starts buzzing right from the start, it plays, and sounds great, but buzzes...I tried different cords, and even different amps, same results..I plugged another of my basses in, and no buzz!!!
I believe you’ve isolated the problem to the instrument itself. Has it always been this way? It would help to at least know the instrument and pickup configuration. If it is a single coil pickup (or two pickups and you’re only using one of them) you will get a hum or buzz like @luser droog mentioned in his comment. This type of hum can be fairly loud but not as much as a live instrument cable unplugged from an amp. If it is that loud then you need to check the internal electronics for dirt and loose connections as well as the jack like @Don Hosek said. If the jack is just dirty some alcohol on a Q-tip might do the trick. If it’s corroded then the corrosion needs to be scraped off or the jack replaced.
You can also pick up interference from lights and other electric and electronic devices if the instrument isn’t properly shielded on the inside. Turn the instrument in different directions and move around the room. If the intensity of the buzz varies adding foil tape inside the cavity might help.