Why does opening AirPort Utility restore my internet connection? My office has a D-Link router hooked up to DSL. Connected to it via Ethernet is an Airport Extreme (6th generation). The Airport Extreme is the wireless router to which my MacBook and other users connect. It's in DHCP and NAT mode, in case that's relevant.

Every once in a while (maybe once every one or two weeks), my internet connection stops working, though the Wi-Fi icon in my menu bar still shows full strength. I used to reflexively disable and enable Wi-Fi, which would reliably fix the problem.

The past few times I wanted to investigate further, so instead of disabling Wi-Fi, I try pinging the Airport Extreme at 192.168.2.1, and I get "Request timeout for icmp_seq". Then I open AirPort Utility. However, the moment I open Airport Utility, everything reliably starts working again, and pings start coming back acked. I'm pretty sure it's the act of opening Airport Utility that's triggering this, as I can wait a variable length of time before opening it. What's going on here?

I'm on an early 2015 13-inch retina MacBook Pro running OS X 10.11.1.
Can you try disabling the DHCP and NAT and putting the device into Bridge Mode? Since the D-Link is functioning as the router there's no reason to double NAT (this is generally considered bad) and that's the only thing that's strange about your setup that could be causing this behavior.