For me, it depends on what's coming up on my riding schedule. If just a normal ride for me, I'm more curious about the "how far will they go" instead of worried about flatting. If a Century ride or other organized ride is coming up or if I were to be going off to areas I'm not familiar with, then I want decent tires under me and will change them before the casing is showing.
I've ridden tires with the casing showing through many a time. I've not had any wreck or scare when they finally ripped apart. However if I was riding mountain switchbacks, it might be a bad experience.