Device says paused but still can access home network

Device (my son's iPhone) states paused on the app, however he can still acess our home network. How can that be? I have turned off the cellular data to his phone and even went as far as suspending the cell service to the phone with no luck. I had my own modem and chose to pay Xfinity the $14 a month rental fee for their modem, so I could have control over his devices.  Seems as I am wasting my money here, if the fetaures don't work as they should.