seems like 5970 uses less power than 6990?
I think in the 'standard' overdrive switch position power draw from a 6990 is about 350W, 450W+ when it's turned on.
But your question was about 5970s, which should come in less than 350W each allowing room for a little overclocking.
Assuming that, with 4 cards you're going to be pulling potentially up to 1400W. That's quite a bit on a single rig - and quite a bit for normal 15A household circuits in the US - which can handle about 1800W, and your wiring will be a bit warm
at that load.
Do you have a model number for that PSU? Looking at ratings for the rails will help us give you a real answer.