In terms of personal experiences, aged / old (hard) tyres on my current Mazda3 caused the rear end to step out twice in a few weeks on wet roundabouts. Fortunately, no one was hurt on either occasion, but it frightened the crap out of me and prompted me to replace my tyres.
During my study for the change, I discovered that the old (OEM fit) tyres did not have a good reputation in general, and particularly as they aged (despite the wear) in the rain. They were still nice in the dry, although a little loud and uncomfortable.
The new tyres had a considerably higher review rating, which I could confirm to throughout their lifespan. I'd suggest rotating tyres so that the fitted rears (on FWD vehicles at least) receive equal use also helps, since wear exposes 'fresh' rubber, which isn't as harsh as the 'outer' surface if left on that axel.
I only modified the second set due to ages (they were 6 years old, but with approximately 4-5mm tread ish if I recall) and because I was reduction in size the alloy tires from 16in to 15in and thus had to change tyres anyway (one was leaking because the alooy was tarnished, and the others appeared to be doing the same). The tyres were otherwise in good condition.