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Grim Fandango refuses to do a damn thing when I click on the .exe file. Graphics card is AI Mobility Radeon 5730. No idea if it supports whatever; don't even know how to find out. Anyone else having trouble, though?
According to this: the ATI Mobility Radeon HD 5730 *should* support OpenGL 4.3, which is more than enough to start the game after the OpenGL 3.3 check.

I would recommend updating your GPU drivers, or even uninstalling and reinstalling them, if updating them doesn't seem to work.