No Connection For A Minute Or Two - DNS Server Not Responding?

I have Broadband Internet and TV through Time Warner Cable. In the past week I suddenly began to lose stations on my cable box from what seemed to be a loss in signal so I had tech support come out to take a look at the problem. Turns out the problem was the wiring in the cable box outside the house. The wiring was over a decade old and had been damaged by water and ice over many winters. Tech support guy replaced all the wiring and the cable box is back to normal!However, after sitting down to use the Internet afterwards my connection kept "dying". Every ten to fifteen minutes or so the connection would seem to die for a minute or two.

All devices connected to the router either wired or wireless experience the issue at the same exact time. Windows diagnosed the problem as "DNS Server Not Responding". Now my router setup has been working for years, and nothing has been changed, so I thought maybe it could be related with the issue with the cable wiresHowever, researching the DNS issue on Google makes me think it's just coincidence.

I rebooted and power cycled the modem and the router. The issue persists. Updated router firmware. The issue persists. I changed my DNS settings to use OpenDNS. The issue persists. Changed DNS settings back to automatic. The issue persists. So I disconnected the router and plugged the modem directly into my main PC and the issue was completely resolved! Hooked up the router again and the issue popped back up after ten minutesRepeated all the troubleshooting steps and the issue persisted through them all.