Most online games use very little bandwidth. 2mbps is more than enough to play any online game. The real factor is the latency between you and the other players. High Latency equates to lag which can really hurt online competitiveness.
Any online speedtest should give you a good indication of your speed, but latency is determined by many more factors, the most important of which is the number of hops bewteen you and the server/peer and the time it takes packets to make it back and forth.