That IS an odd experience. It is possible there was some sort of short in the card (maybe a solder bridge or something similar) that was pulling lots of current and heated up over time (thus the strange bad video card beeps, lack of display, and even the crazy fan speed) then finally burned itself open, thus allowing the card to work again. Its kinda a long shot, but its possible. Have you checked your temps and GPU-z to see what it says about your card?
No, didn't think to check the temps, but the fan was silent, even after a couple hours of fallout, so I doubt there's anything out of the ordinary in that respect.
I was thinking it might be possible that the burnt smell came from the fan...maybe it didn't spin up the first time, and the 30 or so seconds of the card running w/out cooling made it hot after the reboot.
I'll do some more tinkering this evening, and keep an eye on it.
The fan was definitely working the last time, I looked.
But come to think of it, the very first time I booted, it seemed to be pushing air out the back, though I didn't look to confirm the fan was spinning.
I'll do some power cycles tonight, and check for temps and fan speed, etc, and see if anything looks out of the ordinary.
Crazy solutions like change memory slot, change pci-e frequency bus, use DVI to VGA adapter, turn off monitor before boot and so on. I´m very dissapointed with my first ATI card. I had only Geforce cards until now and NEVER, NEVER had any problem like that before. My card works on other computer but not in my new i3... and nor for a lot of other people. ATI sucks! Nevermore...
Actually, thats nothing like the problem the OP had. Every manufacturer has problems sometimes, and its worth doing a little research before buying a card that has known compatibility issues. I've had a number of different Nvidia cards and ATI cards, and have had problems with both, and great experiences with both. As of late, I've had no issues with my ATI cards (4870, 4860 and 2x5750 crossfire), but ended up returning the GT240. Those things just happen. Everything has a failure sometimes and has compatibility issues sometimes- you just gotta work through the issues. Don't just slam a company because you had one problem. Also- whats the issue with it and your i3? Maybe we can help you.