No. Get the video card in your price range. Even the 5770 is adequate for most games. Get one with an hdmi port if you like to run blueray. Your power supply may be fine. I use an ocz 400w with my low end video card, and it's quiet. If you go with the high end video, get a 750w ps by corsair, pc power and coolling, antec, seasonic, enermax, or ocz. Cheaper brands may fail under load or give you bluescreens or artifacts when running games.
Plug your whole system in to the calculator above see where you stand. Don't forget fans, external USB and drop downs for stuff like capacitor aging.
For the card, pick from the list below according to your budget. The one category that essentially no longer exists is the $155 as nothing there is ls less than $175 (excluding off brands) or so now. At $130, the 5770 ain't a bad buy....the 4870 should be about $150 and the 160 is OK at $155. At 1920 x 1200 or above, DX11 is not an issue at this level as neither card is capable of doing 30 fps DX11.
At this point in time, the ones in bold seem to me as the best choices for their price range