Not necessarily, a lot of bad PSUs won't deliver what they claim.
For instance I made some quick search and Diablotech have terrible reputation, so I highly doubt it will deliver the 500W recommended by AMD for a 7950 (or even that 380W you're asking). Some power supplies can have very bad power fluctuations which could damage the card. It's really not a good idea to get a cheap PSU if you plan to match it with high end hardware like a 7950.
That being said Kingwin reputation looks average, not the best but not the worst. So that 750W should get the job done. However, it could be simply defective and not delivering enough power or have already damaged the card. And the other Diablotech is probably too crappy to run a 7950.
It's hard to tell, it could also be the 7950 that failed or that already had a defect like almost DOA. But if the whole computer shut down it's probably indicating a lack of power. Do you have any friends with a better PSU in which you could test the card in a third system?
If you have money to get a 7950, I would suggest you invest more in your PSU in the future. Get a good brand like Antec, Corsair or Seasonic. For instance, I have a Corsair 650TX v2 (650W) and I know it will deliver what it claims and be more stable, I feel safer than with a 850W "no name" brand.