On AC or USB?
USB will not charge a dead Droid.
Yes and no.
I've seen the opposite be true, too. When the battery is completely dead and you plug it into AC, it will start to boot, but the battery will die before it gets all the way booted and is able to charge. It'll get stuck in a boot loop without being able to charge further.
In that case, you plug it into USB with it OFF (USB from a computer, not just from a power brick). The white light will come on but the droid won't boot up automatically. Just leave it sitting there for a few hours. It takes a long time to charge, but once it has enough it will boot up on its own