OK, so the facts are:
1) It drops all ssh connections after 3 minutes. I have never had this problem with any previous router.
2) It drops all VPN connections after 3 minutes. This includes different VPN servers with different configuration info. These also worked perfectly with all previous routers.
3) My cable modem is just an ordinary Motorola modem, not a router.
The fact that the problem occurs for ssh as well as VPN, and occurs for different VPN servers, makes me think it is more than just some VPN configuration issue.