IRQ clash would be unlikely if modern cards are used, I would not rule it out but would expect a BSOD rather than a reset.
I have seen this with a HDD on its way out whereby it hangs for around 3-4 mins then just reboots but that was on Vista on an old old machine of mine which could really have been something else.
Before that I would check also the memory being seated correctly and the contacts not oxidised maybe swap round the memory modules. I can only think that what ever it is (if not power related) is happening at a kernal level and is not being trapped by the OS - so is hardware related in some respect.
If you have an onboard gfx controller on your motherboard and you still suspect the ati card remove it and run off the motherboard for a while, if its only sage etc then you should not really notice but it might give you an indication if the card is faulty.