A 5970 uses 300w, a 5870 about 175w. Total, that's 950 watts of +12V power, about 79A. You'd be running the power supply near it's max, considering the CPU, fans and other stuff that need power too.
The TPQ-1200 is more in line with these power requirements, here's a review
Note the load tests on that site, where they test it with +12V 90A, (and 10A on 3.3V and 5V) = 1200W, also +12V 98A (and 1A 3.3V and 5V) = 1200W. It doesn't go up in smoke, many other crap (ahem..Thermaltake) power supplies die when that site tests if they can deliver rated power.
For what happens when you run a power supply near max, look at the noise graphs of the TPQ-850 here
, the more you push a supply near max output, the more line noise and ripple you get, more likely to make your system crashy.