Wow...I can't think of what it might be; your computer is plenty fast enough to run TPM. I'm not a programmer, and I've never heard of this bug before, but I'll throw out a few suggestions.
One thought...what version of DirectX are you running - 6 or 7? If you're running 7, you might want to download the patch. The patch will NOT help gameplay; it fixes a hardware configuration in the game launcher for DX7.
Also, what resolution are you running the game in? If it's anything above 640x480, I suggest you switch to 640x480 mode until the end of the level; this is just a shot in the dark - I doubt if it will help.
Does it happen when a certain event happens? Or is it when you see someone onscreen for the first time?