I live in a 3 story condo building, and I am on the 2nd floor. My neighbor told me he could hear dripping down below. It only hear it though when you fill up the tub and then drain it? You can't hear dripping when you just let the water run. What could this be? :)

When you fill the tub, do you fill it and then get in it, causing some of the water to flow out the overflow?

Is that neighbor below you or next to you? Does the unit below you report any staining on ceiling or wet spots. Likely a leak in your drain line or trap. Many condo units allow access to the tubs from a closet or other room, does yours?

If no leaks when showering, only when filling the tub, I would agree with Harold. Sounds like the overflow may be leaking. First thing to try is to tighten the screws that hold the overflow plate on. If this doesn't cure the problem, you may need to gain access to the back side of the tub to see if you need a new overflow gasket, or to see where the leak is originating from.