You really only need the speed/bandwidth to download patches or the games themselves. Some games load extra data..ie a bunch of graphic overlays...when you first log in.
Still one you get going the game really is only sending position data for you and other players. It does not send the actual images of the stuff you see on the screen. It just send some command like display object 1234 and you computer goes and find the picture of object 1234 on your hard drive where the game is installed and displays it.
Games are generally designed to allow even people with very slow connection to play. The game company wants as many customers as possible so it would be silly on their part to require people to have 100m connection to play...