A couple thoughts on what this could be, (this is assuming your RCN box is a multiport router, which I've never seen. you didn't specify if you had a RCN box > router > Xbox, or just a RCN box > xbox setup)
1 - It's possible but highly unlikely the ethernet cord is bad. They can carry electricity, which could cause the box to reset.
2 - Try resetting the RCN box to factory default, it shouldn't require configuration.
3 - Try a diff ethernet port, you've probably done this.
4 - It's possible for routers/modems to overheat or not accept connections based on how many devices are connected. So they will work fine if 2 devices are connected, but introduce a 3rd and shit hits the fan. I would reset the RCN box, leave it set for a good 10/15 mins off, when you plug it back in only connect the Xbox and nothing else, if it still has the issue it's not an overload problem.
5 - Call RCN and have the box replaced. It shouldn't cost you money. When they say it's working fine insist it will just keep cutting out every 15 or 30 mins so they'll just get a modem replacement over with. Play dumb, don't get into the technical details of the Xbox with them.
6 - Try your Xbox on someone else's setup if you can. This will tell you if the ethernet port is doing some crazy things or not. If it causes problems elsewhere I'm not sure what you can do except get a wireless adaptor. (trendnet wireless N gaming adaptors happen to be $25 new right now. check slickdeals)
Good luck sounds like one of those really weird issues that may be a bit frustrating to resolve. Number 6 is going to be the real test.
