That does sound odd, and my apologies for the miscommunication. I just assumed you were using DSL
Of course, you've clarified quite a bit, and it seems fairly clear that the issue is in the router somehow.
Are you using a wireless router? We had a wireless one for a bit that issues with heat. Had to make sure it was well ventilated, in an open area, or it would overheat and drop connections.
Had you tried contacting the router manufacturer? They may have some points to help you out, or give youa direction to look for.
Finally, are there any common threads when your connection drops? Any particular software that's running (perhaps something in the background), or a particular household appliance that starts up, or whatever. Any correclation to time of day? All of this info can be used to identify or exclude various failure modes.
Two final suggestions:
Have you tried swapping out the physical cable between router and modem? It sounds like you have, but that's an area I always forget to check until I've wasted a few hours
Second, have you checked the configuration and settings on the cable modem? Not sure if it works like DSL, but our DSL modem also has the capability of acting like a router (including NAT). You can set it up to NAT (modem is the router), shared IP (the IP of the modem is given to the device behind the modem, but the modem maintains the connection), or pass-through (the connection is initiated and maintained by the device behind the modem). We had a bit of problem with disagreement between the mode of the modem and the mode of the router. We initially had the modem using NAT, which ran into my firewall, and the firewall serviced the network. We had some connection problems with this. We ended up having the modem to a passthrough, giving it's IP to our firewall, and that's worked much better.