In 2014, back when the  iPhone 5 was the hot tech of the moment, an image of Rihanna began circulating online. The singer had been photographed walking out of a bar in New York talking on a basic flip phone. It struck a chord: To see one of the most famous—and arguably coolest—women in the world rejecting constant internet connection gave the dumb phone, as they’ve become known, cachet. 

The internet-less phone never went out of production, and in recent years, as smartphones have encroached ever more on our time, sales of old-school phones have risen in tandem, driven by a consumer desire to be less dependent on apps, 4G and Wi-Fi. The prospect might initially elicit alarm (being off-line while out of the house is becoming less and less feasible), but the rewards are unarguably...