Not trying to be a jerk, but I didn't know you could get cable internet that was that slow. I get more than that off my cell phone. Which of their plans are you on? I see they've got 5/15/30/55 in most areas. If you're paying for anything above 5 and you're only getting 6, I'd complain to them.
But, I think you may be over estimating how much bandwidth you have. 6mbps is almost nothing these days. I use 12mbps or more from Steam and Battle.net downloads. A single Netflix stream hovers around 5mbps. Meaning both of those will use 100% of your available bandwidth. That's only one user. Have a roommate/family/etc, yeah, just multiply that by the number of people. Guess what happens when a connection is overutilized? It runs like crap, in which case you have 3 options.
1) Get a faster connection.
2) Use less data.
3) Setup QoS on your router.
The router is easy enough to rule out. Plug your computer directly into the modem.