AirPort Extreme doesn't like more than four devices
I have an Airport Extreme that I've used for a quite a few years as my wireless router with several different DSL (and now Midcontinent cable) IPs. On its network are a two-year-old iMac, a pair of mid-2012 Airs, and and iPhone 6 that's basically the home phone. However, I also have another iPhone 6 and a newish iPad that I occasionally bring home, and while they connected to wireless without any problems, they never get an internet connection. When I take them anywhere else, they connect to wireless and the internet just fine. At home, though, I get nothing but the gray spinner in Wi-Fi next to the network name.
When this started a few months ago, I could Reset Network Settings in iOS and connect, but no longer. I've gone so far to reset all settings on the iPhone (what I'm currently working with) and to hard reset the AirPort Extreme without any change. I'm about to remove the AirPort altogether and try it just with MIdco's Arris Touchstone 8200 wireless router, but at that point I might as well send the AirPort to recycling. What else should I try before doing that? Everything's on the latest OS and the firmware on the AirPort is up to date.