Let me make a stab at this. The reason for the 24 pin connector is that the current draw on sone new motherboards with PCIe and those current hungry video cards exceeds the capacity of the connectors themselves. So they duplicated some of the wires and connectors and remerged them in the motherboard. If your current draw doesn't exceed the official capacity by much, then it probably won't make much difference. This is also why those silly adaptors are a complete waste. They just move the problem from one end of the adaptor to the other.
Besides, if your PSU only has a 20-pin connector, it probably can't output enough current to overload the connector. Aahh, but that's likely your problem - an inadequate PSU. Your PSU is saying, "Hey, for my meager output a 20-pin connector is fine" and your motherboard is saying, "you'll probably need so much current with that video card you're going to get, I'll need 24-pins to handle the power."
Or maybe they sent you another bad video card or maybe it's cosmic rays or something we haven't thought of yet.
Good luck!