When you receive an incoming phone call, the network's towers don't actually broadcast an announcement. What REALLY happens is, every ~3 seconds, your phone connects to a tower and metaphorically asks, "is there anything you want to tell me?". The response has a ~160 byte payload. If you have an incoming phone call, the response includes the caller ID info and furnishes info the phone will need to directly establish a connection and accept the incoming phone call. If you have a SMS text message, the payload includes the message itself (with later extensions for things like longer messages, multimedia, etc).
When you send and receive multimedia messages (including services like Google Hangouts), the bulk of your content actually gets uploaded to a web server, and SMS is only used to deliver a notification that the message exists so the recipient's phone then knows to go and fetch the REST of the message via https.
It's ABSOLUTELY possible to end up in a situation where there are no voice circuits available for a tower site, or a situation where internet connectivity is basically nonexistent. Nevertheless, a true plain-text SMS message will still get through... both sending and receiving... because almost by definition, it piggybacks along with the request when the phone is polling the tower and the response when it gets the answer.
Moral of the story: if you have to get a message through and the cellular network appears to be near collapse, figure out how to configure your phone to send a pure, plain-text SMS message. 99.9% of the time, it'll get through, even when nothing else will.
Tip #2: if you're someplace where there's a strong signal, but seemingly no data connectivity, go into your phone's settings and try to force it to use either non-LTE 4G (eg, HSPA+), 3G (WCDMA/UMTS), 2.5G (EDGE), or 2G (GPRS/CDMA2000-1xRTT). Some towers (esp. Sprint) are configured with dedicated backhaul bandwidth for older services that isn't shared with newer ones. The old services are slow as hell, but nowadays, are almost completely unused, so if YOU forcibly switch to one, you might very well have that service on the tower all to yourself, even when LTE is melting down.
Tip #3: after a disaster, it's entirely possible that your phone will connect to a tower site that has a strong signal, but no actual internet connectivity. Basically, the equivalent of a wifi access point that's plugged in to the wall outlet, but either disconnected from the cable/dsl modem, or the modem itself is unplugged/broken/dysfunctional. It's one of the serious design fuckups of LTE... it fails to account for this exact scenario. Another quirk of LTE, compared to 3G: 3G does "soft" tower handoffs... as you move around, your phone automatically connects to the tower with the best available signal. LTE does "hard" hand-offs... your phone connects to the best-available tower... but then STAYS connected to that tower as you continue to move until it can no longer maintain the connection, even if a better tower is nearby. If you find yourself connected to a dysfunctional LTE tower, toggle your phone's radio (the easiest way is usually by entering and exiting airplane mode), which will force disconnection, then connection to a potentially better tower if one is nearby.