It's been a while since I was an electronics guy.

I believe this depends on your power supply. For example say you have
your 12v/grd line powering something that is 5A constant then you split
it off to another circuit drawing 5A. You're now drawing 10A off of
that single 12V/grd connector. If your power supply can handle it,
you should be okay. If the wire is a small AWG (24awg for example)
then it can get warm/hot (melt?) compared to a bigger 16 gauge (awg)
wire and PSU.

My ASUS Motherboard states that if I am running a 6800GT then my power
supply needs to be at least 400watt psu. Dual 6800's would require at
least 500watts. Anything less and the system will become flaky. Of
course, it has standard molex connectors for each PCI-x 16 slot. I
believe that means I can power it from this extra 4 pins, or from the
molex connector, or directly from the power connector on the PCI-e vga
card if it has one.

I'm not an expert on this subject. You said you weren't running PCIe
so it probably doesn't matter.