iPhone SE 3G connection won't access internet or send iMessages

My new iPhone SE won't access the internet or send iMessages when it's in 3G mode. It says I have no connection to the web and must enable iMessages (which is already enabled). My wife's identical iPhone SE (bought one day after mine) has no such problem. We've checked every setting we can think of and they are identically set up. 4G works fine on both phones (as does WiFi), though for some reason both phones will randomly shift back and forth between 4G and 3G, even when we're stationary, like sitting in a restaurant or on a park bench. We've tried restarting my phone, but no change. Any insights, anyone?