From my experience it isn't so much about adding warmth as it is protection. With a small tarp it'll protect from spray and the bottom is a ground cloth. It creates a layer of still air around the bag/quilt so it doesn't loose heat to moving air as easily.
I don't think it adds warmth per se. Bags are often rated for 'in-tent' conditions so the bivy makes up for the lack of coverage of tarp/cowboy camping.
I use a bivy for bug control and often for-go the insulation layer... but that's only when I'm out near the coast.