In this era there were lots of competing APIs like Glide, S3 MeTaL, D3D, OpenGL. So you could end up with games that only supported particular APIs and not the one(s) supported by your GPU. At the time D3D and OpenGL were kind of a lowest common denominator, unless the game happened to support your particular vendor's OpenGL extensions (if they had any).
From memory unreal tournament and other fps that used the same engine had support for opengl, glide, d3d , s3tc and software rendering. It was one of the most compatible render engine
I remember some earlier titles that were locked to specific cards such as the Matrox ones and didn't support any other accelerators.