My mistake, the supply is rated at only 600W not 650 W. It is a CoolerMaster RS-600-ASAA. It has 38 A total for the 12V bus listed as the "Continuity (A). Let's assume they meant "Continuous (A)". The 38A comprises three sections - 16A, 14A and 8A.
Maybe the problem is I have overloaded the 12V section I am using. Perhaps I should feed each card from a different 12V section (via two different harnesses) rather than both from one harness (which I did to simplify the wiring).
The link above takes you to the product description on the CoolerMaster web site. I believe it is a decent power supply and is not "pitiful". You just have to be careful not to compare peak output ratings with continuous ouput ratings. Peak output of the supply is 58A on the 12V bus.
So, I had this power supply sitting on my workbench and installed it last night. This beast can source 98A on the 12V bus. It runs very quietly, which I love. I still have both cards on the same 12V bus (dumb!). I'll rectify that tonight.