I once mined with a 5850 with an old 330W PSU for about 10 hours before it failed. The PC would kind of turn on (fans spinning), but nothing would happen. I replaced the PSU and everything worked fine.
If you have a weaker video card available, give it a try, and use the secondary PSU to power the whole system. If it works, you know the problem is either the 5970 or the PSU (more likely?).
I have four 5970s actually, two were damaged this way. It's so bad... I sometime hotplug my 5830s to PSU and PCI-E and they are still working very well...
Whoa, hold on a second. You have 4 5970s and you already damaged two of them because of an inadequate power setup?!?! Dude, do yourself a favor and get a decend power supply ~ 1000W and up!!! Stop killing the 5970s -it's a crime and you should be arrested lol