Thanks all!
Believe me, I've been thinking about your blown tire picture a lot since you posted it, magohn. The tread pattern on my tires is the same as the one you showed us, but the model is different. Mine are Specialized All Conditions, which came with the bike. I've been inspecting the tires after every ride, and so far they haven't shown any signs of failing. The slight cracking in the sidewall above the tire bead has always concerned me, but it hasn't gotten any worse. It may have been like that since the bike was new, but I'm not positive.
I'll probably replace them this season, just to be on the safe side.