What you described to me looks like some sort of safety system in the power supply.
I recommend that you disassemble whole computer and check for any burn marks on the motherboard and pretty much everything else. There could be something which causes a short and makes power supply turn off. Pay close attention to computer case and check if it's moving or vibrating.
Next step would be to check cables in your power supply. If you have multi-rail power supply, you could be hitting rail limit for power. Make sure that load is distributed among cables and devices. If a cable has several connectors on it, try to use up all other cables first before using extra connectors. Make sure that you balance load on cables with several connectors. For example on a cable going into a hard drive make sure you only connect low power consumers and so on. This in general shouldn't be a problem, but some power supplies can be picky and may have bad load distribution.
Next check how different devices are mounted. I once had a similar problem. I bought a new computer and every once in a while it would shut down unexpectedly. In the end it turned out that the incompetent or malicious manufacturer used too long screw on the floppy drive. They penetrated inside and caused shorts from time to time. Make sure that nothing similar is happening in your case.
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!
Best Answer
Here is a simplified explanation of what's going on in the power supply. The power supply divides the input voltage (230V in your case) by a fixed amount to give the voltages needed by the motherboard and other components. There are various outputs, but one of them is 5V. To get this you have to divide 230 by 46, or divide 110 by 22.
Power supplies can work on 230V (most of the world) or 110V (North America). The switch selects which divisor is used inside the power supply. In your case you set the switch to 110. This selected the 22 divisor. However, you applied 230V to the input. This resulted in a voltage of about 10.5V on the 5V circuits. This voltage was too high and something blew. It may not have been on the 5V circuit, but the story is still the same. You applied too high a voltage for the setting of the divisor. If you are lucky only the power supply was damaged and all the other components are still OK.
If that didn't make sense, then consider this. Suppose your car is stuck in the snow (I'm from Canada; somebody's car is always stuck in the snow). If you push the car out by hand you won't damage it. If you push the car out with a tractor you will probably dent the metal. The tractor pushes too hard. The voltage is like that push. Voltage pushes electrons around. When you applied 230 volts to a circuit designed for 110 volts, you pushed too hard.