Will backup cell system work if internet cable is cut?

If a thief were to cut the cable coming into my house that provides my internet service, would the back-up cell system send any alarms to dispatch?  In this case, the wifi router would still be powered on, and the base station would still be connected to it, but it would have no communication to the outside world. Does the base station in any way verify receipt of the alarm signal with the monitoring service and switch to the cellular if it gets no reply?
The reason I ask is that my ISP recently went down (the problem was for sure on their end), but I received no notification whatsoever from the armed simplisafe system.  When I tried to connect to it via the app, it was "offline".  It obviously was not connected to the cellular backup (I presume because it still detected wifi - the router was up, just not connected to the internet).  I am 1000 miles away, so this was a little disconcerting.  I can probably test this myself when I get home, but it seems the backup cell system is of little use if a thief could just snip the coax running into your house.