There's way too many reasons you could have laggy games. I'll list some
- High speed internet doesn't equal less lag. You can have a 2mbps connection and have amazing pings and low latency. It's not the size of the pipe, it's how it goes through the pipes. You can have faster internet and lag more. Why? Because going from point A to point B is the same speed, having more bandwidth (aka faster internet) just means more can go to and from, nothing goes faster.
- Other people use your internet - People who live alone only have to worry about downloading while they play, but in other homes you have to worry about other users. Someone could be watching netflix while you play, and if your router is average, it's probably not helping. Some routers will give priority to specific data, others don't care.
- Sometimes your ISP sucks - this isn't about slow speeds, it's about routing, packet loss, and jitter. During peak traffic times, sometimes your data is dropped, or router a longer way. This can result in lag and delays, the unfortunate part is there's nothing that can be done. Sometimes your connection won't be stable, sometimes ISP oversell and certain neighborhoods can have slow speeds and bad packet loss. Sometimes data that takes 10ms to get to a server will take 50ms, resulting in more delay (jitter).
There's tools you can use to help you determine issues.
Ping Test -
http://www.pingtest.net/ - Try different areas, opposite side of the country, etc.
Checking Routing Problems -
http://www.pingplotter.com/freeware.html - Visually shows the times between routes, can show if you're lagging locally or if it's with your ISP
Just keep in mind, speed (mbps) has nothing to do with lag. Faster internet does not mean faster packets, less lag, or less packet loss.