In case it helps anyone, I've just had a 4S with this problem. Completely dead screen but when plugged in to a charger it does the double vibrate thing like on start up every 4 seconds or so but doesn't actually turn on. Also tried a hard reset but that didn't do anything. What fixed it for me was disconnecting the screen connectors at the top of the logic board for a minute or two (dont know if leaving it disconnect for any length of time helped or not, I just went to make a coffee) then reconnected them, then I either just turned it on as normal with just the top button or I did a hard reset, I cant really remember but I did one of the 2 and it turned on. Hope this helps someone.