I'm not sure which kind of 5-pin jack your '95 bass has, but it's not that unusual for there to be a little bit of motion of the plate that the pins are mounted on relative to the shell on any Cannon-type connector. The theory here is that the pins can shift as a group when mating with the plug, otherwise everything would have to align perfectly to make the connection. Some jacks have more wiggle than others. The relationship of the pins relative to each other must be absolutely stable.
An interesting test for you to run would be to see how things sound when you connect on the 1/4 jack. By '95, the default wiring was that the 1/4 jack is mono output (e.g, you can hear both pickups with a regular 2-conductor guitar cable). On instruments from the 80's the 1/4 jack is usually stereo, so if you plug in a regular guitar cable, you'll only hear one pickup, usually the bridge one. To get both pickups on these older instruments, you need a 3-conductor 1/4 plug to two 2-conductor jack y-cable.
If your bridge pickup sounds OK on the 1/4 cable, then I'd start suspecting the 5-pin cable, DS boxes or the 5-pin jack. Cables will almost always be a more likely cause of failure than the connectors since the internal connections bear the brunt of you walking a foot too far from your amp.
You haven't talked about the specifics of your amp setup yet, but if you have a stereo rig with the pickups routed differently, you should check those paths as well. When you switched the bridge and neck pickup leads you mentioned that the bass didn't behave differently, but that answer was ambiguous. Did that mean that the bridge pickup continued to be the noisy one or did you mean that the neck pickup became the noisy one? If you have a stereo amp setup, and the bridge pickup really had a problem, you'd expect the noisy sound to be coming from the speaker that the neck pickup normally plays through without noise. If you swapped leads and the noisy signal stayed in the same channel, then it's not actually the pickup that's the problem. If you have a mono setup where you take the mono out from the power supply, then diagnosing by swapping is a little harder to detect (you need to gently tap the pickup face with a screwdriver to identify which pickup is which).
If you poke around in the electronics while the amp is active, it's not necessarily odd to get thumps or noise when you move the wiring (in certain cases, the wiring will seem to be microphonic). If you poke around while the amp is on and something completely cuts out or radically drops in level when you move a wire around, that would be a sign of a bad connection. When you touch the jack pins and get hum, that would be normal; again, if the sound cuts out entirely that would be an indicator of a bad connection.
Try this stuff out and report back. You've seen what the internal wiring looks like - the Series bass easily has the most complicated 1/4 jack wiring of any instrument in existence, so it would neither be fun nor cheap (I guess I mean that in absolute rather than relative terms!) to try to replace it.
David Fung