I have a fiberglass tub surround on the second floor and when the water reaches the overflow it leaks onto the ceiling below (now the floor since the ceiling had to be removed). I've tightened the screws on the overflow cover and even applied a bead of plumbers putty behind it but no luck. I noticed that there is a thick black gasket behind the fiberglass which I suspect is the problem. Are these prone to wear out?

So you've visually confirmed that the water is coming out near where the overflow line connects to the tub?

Yes, that washer/gasket can "wear out". Not "wear out", really, but since they are rarely used, they can get dry rot.

But also double check that the overflow's connection to the drain is good too; that's the one other point that the water could be coming from.

I have a fiberglass tub surround on the second floor and when the water reaches the overflow it leaks onto the ceiling below (now the floor since the ceiling had to be removed). I've tightened the screws on the overflow cover and even applied a bead of plumbers putty behind it but no luck. I noticed that there is a thick black gasket behind the fiberglass which I suspect is the problem. Are these prone to wear out?

I can't add much to the excellent answer you got from Rick, he was right on target. While the spongy gasket behind the overflow can "dry rot" and deteriorate they can also compress over time and began to leak.
But my biggest question is why would you fill the tub to up over the overflow?
A really large rubber ducky that needs more water to move about in? Doing laps around the tub? Submarine war games in the tub? Or just wanted to see if the overflow was really water tight? Inquiring minds want to know! Cheers, Tom

When making gin in the bathtub, be sure not lose any of it out of the overflow tube :p