Curious “server DNS address could not be found” problem

This is a problem that occurs only when I use WiFi in hotels. I’ve been in two motels in New Mexico this week, and the same thing has been happening.\. What happens is I’m usually able to access my website (route66news.com) and go to the admin page. But after a few minutes, this error page pops up, and I can’t log in or do anything. The website continues to work, as I can access it with my phone, and everyone else can access the website, too.

Is it some sort of WiFi security setting that’s dialed up too high, or some other subtle problem on my end? Like I said, this seems to happen only in motel WiFi systems.