Twilio extension stopped working; neither individual nor mass SMS go through
Matthew Wire nailed it: the most current version of the Twilio extension for CiviCRM doesn't work with PHP 7.0 -- but luckily for us, a huge patch became available through GitHub just one week ago. Since we need PHP 7.0 for other stuff on our site to work, reverting to 5.6 wasn't a viable option.

As soon as the updated Twilio SDK extension was in place, I was able to run those scheduled jobs manually again. This experience also showed me how just one scheduled job that doesn't complete can derail the CRON train altogether. This gives me at least one idea for troubleshooting a persistent CRON headache in which our scheduled jobs run some of the time, yet we haven't yet isolated the problem. It's all connected, isn't it?