My bridge pickup gives a very, very low volume compared to the neck pickup......it used to be equal........what could be the cause??

If it is a recent 4003, you have the "vintage tone" push-pull switch on the bridge-pickup tone control. Try that and see if you have it set in the vintage position which cuts bass response on that pickup. Just guessin'.

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Author: "The Rickenbacker Electric Bass - 50 Years as Rock's Bottom"

It's a 2003.....so no push pull.

Is it possible to have a broken pickup but still have some sound from it??

Yes. Inductance. The way to find out if it is the pickup is to unsolder the "hot" lead of the pickup and measure the d.c. resistance of the pickup to ground. We aren't looking for any particular number, just that the value is stable and similar to the other pickup. If the number is similar to the other pickup and stays stable, then the problem is elsewhere, possibly a solder joint. But if the value varies to the point of never settling down, or is very high, or doesn't read at all, then yes, it may be a break in the pickup windings. It is possible, but most unlikely, that if you have the .0047 inline capacitor to the bridge pickup in the circuit that the capacitor opened up. But this essentially never happens. To check the capacitor, temporarily bridge its leads and see what happens to the signal. Ideally, when bridged, the tone should have about the same output, with just a little more bass.