|To get an idea of the maximum size textures your graphics card supports, you can use "proxy textures" in OpenGL 1.1 and above.|
This can depend on the specific implementation of your textures, which is why this can't always be known in advance.
Note that your graphics card may support larger textures, but there's no way (that I know of) to tell if it actually uses hardware acceleration at those levels.
But at least you know if can physically do it without crashing.
Example output for my four-year old laptop with Intel Graphics:
Example output on my four-year old PC with a nice shiny GTX 460: