Are your windows wood or plastic? Are they older and leaking air? Sometimes with older windows cold air will leak in and hit the warm air in your house and cause moisture. This may be the cause.

You may also want to purchase a humidity monitor and check the humidity levels in your house they are about $10 or so:
http://www.amazon.com/Acu-Rite-Indoor-Humidity-Monitor/dp/B0013BKDO8/ref=sr_1_1?ie=UTF8&qid=1327686215&sr=8-1

I hope this helps.

Thanks for your help, Madalyne

It could be mold or it might not be. If you are seeing black spots and dampness you first should determine where the moisture is coming from. Is there a roof leak or maybe are their pipes running above your room etc..

If you are comfortable removing the mold yourself ensure you have the proper safety gear. Tape off the doorway with plastic and open a window(if you have one) in the bathroom. You can then scrape the mold off the 2X4 or use a rotary tool (often attaches to a drill) to take off the mold/mildew. It is important to have proper safety gear because this basically sends the mold airborne. After the 2X4 has been cleaned and all visible traces of mold and dust have been removed you can apply a sealer such as Kilz before closing up the wall. It is important to read the directions for Kilz and follow them completely.

I hope this helps. If you feel it is too much work, too big of an area or are uncomfortable doing the work to remove the mold call in a professional. As always do additional research on the above advice(and use at your own risk) as it is often hard to give advice on something without seeing it in person.

We are having our main bathroom renovated and everything is being ripped out. Although there was very little or no evidence of mold/mildew on the tub tiles, now that they are out, there is one 2×4 stud that has black mold(?) all along its length on 2 of 3 exposed sides. What do you recommend?

I have never heard of mold entering a water supply passing through the pipes. I would still be concerned with the mold though, I assume the water that enters the pump room is really cold(prob well water), the air in the pump room is probably warmer which causes the pipes to sweat. This moisture is most likely causing the mold. They sell pipe insulation that should help with that. Also consider cutting out the drywall and replacing it with green or blue drywall which is more moisture/mold resistant.

Thanks!

You said you still get water when there is heavy rainfall. Where are you getting this water? In your basement? Have you had someone come out and look at the grade of your yard behind your house? Are there any areas of your house that are below grade(even slightly)?