You could, but that would give you two air gaps and three leaves, which will very likely make the ceiling even LESS soundproof than it is now. The same concepts apply as with walls.
If you don't want to remove the existing ceiling panels, you'd be better off just adding two more layers of sheet rock with staggered seams.
Part of this depends on how much noise you get/give from upstairs, and how much it matters. If it's critical, then you should remove the ceiling panels, put two layers of tightly fitted sheet rock between the joists and tight against the upper floor, caulk well and support with cleats, then add at least 3" of 3 lb/cu ft rockwool insulation, then resilient channel screwed to the bottoms of the joists and two layers of 5/8" sheet rock over that. Build everything like it was going to be a boat and you didn't want it to leak. Use the expensive silicone caulk at the very least, or try to find acoustic-rated caulk locally if possible.
Doing that should get you close to 60 dB of sound attenuation between upstairs and down, provided there are no "sneak paths" for the sound to travel along. The only difference between sound leaks and water leaks, is that water doesn't usually leak uphill... Steve