Low-bandwidth is not the only advantage of text messages, although that's a huge one. As Thseng pointed out, store-and-forward let's a busy network queue up messages and they can squirt through when the network is available.

The other reason why text messages often sail through while voice calls have overloaded the nearest cell tower is because (at least with GSM) text messages are transmitted on a signalling channel, not the main voice channel. The singalling channel carries info like telling you that a voicemail has arrived, so it would never be jammed with voice traffic.

I have no idea if bandwidth-intensive uses like the newer video messages or ring tones are sent through the signalling channel, though. If they did, then thousands of people sending videos of their destroyed homes or whatever at the same time would easily block the ability to send text messages, too. I don't remember, but I don't think video messages were really around yet during 9/11. That would be the kind of situation where a lot of people could've been trying to send video messages at the same time in a small location. The London train/bus bombings was definitely in the age of video messages, but I never read anything about video messages clogging the system, though, although I read that voice calls were jammed. Maybe video messages are not really something to worry about during a disaster. Still, I'm sure they eat up your phone batteries to send and play back.