there seems to be a lot of people asking about running 5770's from low powered PSU's. Here is the link yet again http://www.tomshardware.com/reviews/geforce-gts-450-gf1... that system is fairly minimal a core i7 with an ssd hard drive. Depending on your components the power draw may be different. Also the power draw across different rails is not covered. I wouldnt trust a 250w psu, it may run, but it may blow up too...Also when you say "industry standard" what does that even mean? what brand/model PSU is it exactly?
There are some low-wattage Seasonic and FSP PSUs that are actually good for more than their labeled wattage, not that running that way for long is a particularly good idea (and I think they're 300W, which would be enough since the +12V rails are strong).
Anything else though, I would not trust in the absence of a competent technical review (e.g. with load testing, waveform analysis, etc).
Assuming the PSU is of even modest quality (e.g. not Bestec), you should have no trouble with a HD5670.