Actually, that depends on the card. Some of the high-end AGP cards (IE X1950's) use a PCI-e power connector, which comes off a 12v rail (unless you are using a standard 4 pin splitter; not recommended).
Most of the time, the system will either reboot randomly or won't power on at all. It's EXTREMELY unlikely (I would say borderline impossible) that it's only going to lose power to the fan since most of your peripherals are running off that rail. If your +5v rail isn't supplying sufficient power your hard drives, cd/dvd drives, and fans will be losing power as well. More over your drivers have temp warnings and most of the newer GPU have a core shutdown feature to prevent it from roasting. nVidia got sued a while back because the core shutdown never activated on a guys card and started a fire. Don't recall the outcome of that.
That is why on newer power supplies the video card(s) have their own 12v rail. Mine has quad 12v rails; 12v 24pin, 12v 4pin (P4), 2x 12v PCI-e.