For a system using two Radeon HD 6970 graphics cards in 2-way CrossFireX mode AMD specifies a minimum of an 800 Watt or greater power supply. The power supply unit should also have a combined +12 Volt continuous current rating of 63 Amps or greater and have at least two 6-pin and two 8-pin PCI-Express supplementary power connectors.
You may find power supplies on the market that supply more than enough Wattage to run the system. However, many of them lack the Amperage capacity on the critical +12 Volt rail, which is necessary to support that claimed Wattage.
The two devices that consume the majority of the power, within the computer, are the CPU and the GPU. Both of these devices obtain all of their power from the +12 Volt rail(s). If this critical rail is unable to supply enough power to these two components the system is not going to run properly.