If a PC game supports a certain video card and you upgrade the video card to a newer version of the same brand will the game work with the new video card

Yes in most cases it will. There are always exceptions though and the older the game is then I guess your likely to get more problems with the newest cards.

If a game doesnt work with your newer card though its worth checking for game patches as these often included updates for newer 3d cards, and indeed its worth checking the tech support forums from the game publisher.