Hi there,
as far as I can see most users experiencing this problem are in the USA. I am over here in Germany and had the same problem. I got the iphone on friday and didn't play or install much with it. Saturday morning it was dropped dead. NO reaction whatsoever I did. I had a nearly hour long call to the apple hotline and we couldn't resolve the poblem. We recieved the message when plugging the phone to the McB that there was a iphone which I should restore, during restoration it said software failure 1600 and broke up with the process. Later that day I went to the Apple Store and also no result until the phone was hanging on their Macbook and the Manager pressed some buttons unintendently. Then it awoke and ran until I went to bed. Next morning same thin, all dead. I could revive it with this: pressing the home button while plugging in the USB cable. Than the restoration message came up, when so I pressed both buttons home and on/off for more than 10 sec, after that the phone startet to reboot itself without any need of restoration. Afterwards all was fine. I changed it anyway because I think this is nothing I should have to do with a brandnew device everytime I don't pay attention to it. On sunday the dead situation occured three times during the day. On Monday and Tuesday morning once after the night. No apps have been running when I put it by side. It is a 16 GB Iphone. Today I got the replacement and can say more about its function tomorrow morning, unitl now it is working fine. As a sidenote, because I got the Iphone directly from Telekom the Apple store couldn't exchange it for one of the plenty phones which were there, because I didn't buy it at there place. For replacements of iphones bought whereever they need an extra supply which hasn't arrived yet ...
So Apple send me one directly with UPS after my second call on sunday.