5 Max connections/devices to the internet? Is my ISP limiting me somehow?

I have a strange limit where I can only have 5 devices connected to the internet at any given time. More than 5 devices can connect to the router, either through WIFI or ethernet, but only 5 can use the internet.

I've tried this on a Buffalo router, and now I have the Apple Airport Extreme. The different routers didn't matter.

Can the ISP somehow limit the number of devices connecting to the internet, even behind a NAT router? Is there a setting I need to change in the Airport Extreme to hide the presence of multiple devices?

My ISP is Nextera Communications according to speedtest.net.

More details: The computers are actually connecting to the router. Just no outside internet. The devices are modern devices, such as iPhones, iPads, Macbooks, and Windows 7 PC's. This has been happening for a long time, and through multiple routers. The limit is definitely fixed at 5 devices.

Solution 1:

If your ISP is limiting you to five devices, double-NAT may be your best solution.