This is definitely a graphics card issue. Since this card is not supported you have two options, buy a card that is supported or try different versions of drivers for your 5870 in the hopes that one will work for your needs.
This probably has to do with the complexity of some of the files, or certain appearances or features in the model that are causing issues.
It is most definitely a deficiency in your video card, and there is no real solution for you, since you have a laptop. There is no way around this, except to go to Tools>Options>System Options>Performance and click the check box for "Use software OpenGL". This will disable your video card completely inside SolidWorks, and your CPU will be doing all the work. Note that this can only be turned on when you do not have any part files open. It also makes SolidWorks run very slow, so I suggest you only turn that on when you have to.