Motherboard won’t power-on until it rests for ~10min with no power plugged in

Most likely it's the power supply.

The video cards can draw a lot of current from the 12V rails. Some of the cheaper power supplies with lower wattage ratings can't do it, especially on a 2 card setup like you have. So I'd start by looking there. First remove all the cards and see if the motherboard powers on (that'll verify a card hasn't come loose and is shorting the pci-x bus).

If not, have a look at one of the many howto's on the web for testing your power supply. e.g. http://pcsupport.about.com/od/toolsofthetrade/f/powersupplytest.htm If you don't have a tester, then open it up and check fuses. See below:

Personally, I think 650W is too low for a 2 card setup (particularly those cards. They are toward the high end of current draw). You should have at least 800W or above. For a 3 card setup, 1200W+ is recommended.

I should add that most power supplies have overload or short circuit protection. See: http://en.wikipedia.org/wiki/Power_supply

Overload protection

Power supplies often have protection from short circuit or overload that could damage the supply or cause a fire. Fuses and circuit breakers are two commonly used mechanisms for overload protection.[6]

A fuse contains a short piece of wire which melts if too much current flows. This effectively disconnects the power supply from its load, and the equipment stops working until the problem that caused the overload is identified and the fuse is replaced. Some power supplies use a very thin wire link soldered in place as a fuse. Fuses in power supply units may be replaceable by the end user, but fuses in consumer equipment may require tools to access and change.

A circuit breaker contains an element that heats, bends and triggers a spring which shuts the circuit down. Once the element cools, and the problem is identified the breaker can be reset and the power restored.

Some PSUs use a thermal cutout buried in the transformer rather than a fuse. The advantage is it allows greater current to be drawn for limited time than the unit can supply continuously. Some such cutouts are self resetting, some are single use only.

If your PSU appears to be dead. Carefully remove the screws (CAUTION! High Voltage - be very careful what you touch). You may see some fuses inside. Are they still intact? If they are not then it's what's likely happened is that too much current was drawn and the fuses popped. Then you can replace the fuse and use the PSU in another machine. Now go out and buy a higher wattage PSU!

It was a failing power supply.

This latest time, it just wouldn't start back up. I tried replacing the CMOS battery per davidgo's answer, but that didn't work either. I found a guide for manually testing a PSU, and with it disconnected from everything but the wall, switched it on with pins 15 and 16 jumped. These are supposed to signal the PSU that it's on and should start spinning its fans, but nothing happened.

I did buy it in January 2006, so it certainly doesn't owe me anything!