According to one link, you're out of luck if you have the following cards:
Blizzard lists the most common cards that will not be able to run StarCraft II. They are NVIDIA GeForce 6150, 6200, 7150, 8300, 8400, 9100, 9300 and ATI Radeon Xpress 200, 1150, 1250, X1050, 8500-9600, X300-600.
My card is stronger than those listed, so wtf?!?!
Heck, I even see this advertised:
MSI GeForce GT 240 1GB w/ Starcraft II Trial $60 at Newegg
I took a snapshot of the BSOD: IRQL_NOT_LESS_OR_EQUAL. Blizzard's site says that means I need to update some drivers.