Well you gave out all the specs, except the ones that count the most: the video cards. No mixing ATI and nVidia right? (sorry, I had to ask) Still it sounds like power starvation to me. Does the pc post with either of the video cards in it? If so, then it just might be an issue with the psu. Did you buy a quality psu? Some of the cheaper models barely pack enough amperage on the 12v rails for one video. If you are sure that none of the video cards are individually responsible for this behavior, you should buy an 870w psu and try that. Check the store's return policy just in case; Best Buy has one of the best return policies around; you should consider buying there.
Also if the board is still under warranty, place a trouble call directly with Intel, just in case they might know something about this situations.
Let us know how it turns out.
Sorry for the late reply. School has been busy. The video cards are both Nvidia. I just got a new PSU. 1000w BFG Tech. It did not solve the problem. The computer still will not boot with both cards in. Do you think that it might be a mobo problem? I am going to try to warranty my mobo next and see if that might be the problem. Any other suggestions would be helpful. Thanks!