As pointed out "ping" is the thing that is the main thing to worry about for online gaming.
I have 2 fibre connections
1) A+A that is 32 down and 7 meg up
2) BT Infinity that is 38down and 9 meg up
My ping on A+A is almost half the amount of the BT one and I also get a constant speed when downloading with them unlike BT which can be vary a lot.
Playing actual game online use next to no bandwidth online so speed isn't really involved at all, looking at my usage yesterday for instance I played BF4 for 4 hours yesterday and the average usage for download was around 200kb's and the upload was less than 100kb.
I used to game on a 1.5/0.4meg connection all the game and had no issues gaming online at all but then I made sure that I joined a decent ISP that allowed me to get the best possible ping etc.
ping/latency is king in games, Unless you have a slowish connection and a lot of people on the same connection doing stuff like watching videos online then speed really shouldn't be an issue for online gaming.
TV's can make a big difference to lag as well especially the newer ones as they have so many things turned on which just add input lag to the picture, You need to make sure that if your TV has a game mode that it's turned on so that as much of the stuff that adds input lag is turned off.