It really depends on what you are doing. For example, Diablo II uses almost no resources as a client. However, last weekend I hosted a 7 person game and it filled up all my 512MB of RAM plus used 600MB of swap (This is with patch 1.02).
I like having 512MB of RAM on my main machine. Diablo II loads acts on this Celeron 566 @ 875 with 512MB in about half the time it does on my next best machine a P3 500 @ 560 with 192MB which for reference loads acts in Diablo II within the same second as a K62-400 with 256MB of RAM. However, the load bar goes at different speeds on each machine at different parts. It seems that both CPU and memory play a part in how fast acts load in Diablo II.
Darkstone which is similar to Diablo II, can benefit from up to 256MB of RAM.
I don't play FPS much, but it doesn't seem to make much difference past 128 or 192MB of RAM on most of them that I have played. Video card and CPU seem to make much more difference.