You should be able to run the 7870 just fine. The problem should not be the PSU itself.
Dell's PSUs aren't all that bad. If it says 460w, then it means that it will be capable of nearly that amount of power. I'm not endorsing Dell.
500w is a RECOMMENDED power wattage. What matters is the amperage that the PSU can deliver; I am not exactly sure the amps and volts specs of the PSU.
A 460w PSU with a 7870 will not cause your computer to not boot up. Something else is not working properly, it should not be the PSU or card unless you have a faulty unit of either.
You mentioned that you reseated the card. Did you connect both power connectors back to the card?
