Hmm, I cannot tell exactly what the cable company has given as a modem, but I've had it for years with no problems.
The router is the big Airport - Extreme. Had it about 4 years and works fine.
I did note that my other wireless devices outside of my desktop had both a wireless signal and connected to the internet, so both router and modem were doing their jobs, it was just the Mac Mini that was not connecting. I used the wireless diagnostic built into the OSX El Capitan to generate the message I copied into the post.
No extender.
As I said, this morning and all the mornings before - no connection issues. This afternoon, I received the message after booting up my mac mini 2014.
It was if settings had been changed, but nothing had been upgraded or changed between the few hours I shut down the system this morning and restarted it again this afternoon.
I did do a full restart (even though I was getting fine wireless internet connection on both my ipod touch and my iphone 6) of the modem and the Airport.
The message ""you are connected to a wi-fi network that cannot communicate with your wireless router (aka Airport)." says to me that I had been switched to another wi-fi network other than my previous one , so i am asking if such a thing is possible.