I looked into it a bit for you...
It turns out that router you have is rather old, and the NAT issue was extremely common on it. The problem is due to the NAT table being very small, something around 5,000 addresses. While that might sound like a large number, it really isn't.
Unfortunately your options are rather limited...
You can contact Verizon and see if they can Bridge the router, giving you a static IP to place on your home router.
Replacing the Verizon router with a more current model.
I highly doubt the first option will work out as Verizon uses DHCP for their residential customers, or in other words, a non-static address.