Lets say the answer is mostly no. Most online games are designed for so people with small internet connections can still play. Things like wow were designed to work on the older dsl connections where you could not even get 500kbps upload speeds. Some newer games take more but I have not seen any that wanted over 1mbps.
The main time games do use huge bandwidth is when you originally download them or it seems lately every time you launch some games and they want to install the latest patch.