If chess and minesweeper don't work then I don't think the graphics card is the problem. Perhaps the drivers are messed up, but then why would BFBC2 work with high settings? Weird.
Perhaps EA finally decided to bring something sneaky into it's games so that your comp will only play EA games. They are greedy.
If ALL ELSE fails I would reformat, but do so as a last option. I would try system restore first.