We are getting water seeping through the broken caulk around the tub which has not been used for at least a week , we found the leakage while painting and having the bathroom out of commission. How could it be leaking with no water running in that room? Could it be built up from the broken caulk? If so how do I get it out from under there?

There should be a tub access hole in the wall behind the front of the tub. Open it up and see if you can track the leak back to the source. The only way we can help is if we know what's leaking. Back to you, Tom