It's a matter of how much power the GPU needs. The socket supplies up to 75w, a 6-pin connector adds another 75w and an 8-pin connector adds 150w. For example the 5670 can use up to @ 63w so it doesn't need an additional connector. The 5770 uses @ 100w so it uses the socket + a 6pin connector. The 5850 uses @ 150w so it gets the socket + 2 - 6 pin connectors. The 5970 uses @ 300w so it gets the socket + a 6 pin and an 8 pin.
Electrical code. The wires on a PSU are small. You send to much power through them and you can literally melt the insulation off. By using more then one set of wires you can safely send the power that the video card needs without burning anything down/out.
I just purchased a hec HP585D 585W ATX12V Power Supply and a XFX HD Radeon HD 5850. I really don't want to buy another PSU, would it be acceptable to use 2 3-pin connectors with an adapter along with a 6 pin connector? Would I damage the GPU or PSU?