Electrical failure, open circuit or ground fault. I'm guessing the mobo is improperly secured and grounded. Or one of your ATX power cables (or even your power switch) has a loose, intermittent connection. This can also be a problem with an all-plastic PC chassis (or even a metal one covered with thick nonconductive powder-coat).
You can reseat the HD6870 into its slot, firmly, and use a proper metal backplate screw. A real metal screw, not some clicky plastic tool-free gimmick.
Your PSU might also need to be firmly (re)secured into the metal chassis. No vibration-damping rubber/gel absorbers or whatever.
You can screw a ground connector (something like
this) onto a motherboard standoff, wired (and somehow affixed) to directly contact your PSU's metal chassis.
Your problem might actually be improperly seated DIMMs. Never hurts to check.
Worst case - your mobo has cracked traces or solder points somewhere. If nothing else works, thoroughly examine every trace and every solder point (on both sides) ... and, even so, any damage or flaws hidden within inner layers cannot be visually detected.
"All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others." - Douglas Adams
[/Korth]