I've had two recent experiences with old tires.
In one case the tires looked brand new. The tires had been place on the vehicle, hardly ever used for 5 years (less than 1000 miles), and then the car sat in a climate controlled garage for 15 years.
Even though the tires had no checking or cracking or even dry rot, the rubber had lost so much of its elasticity, that they handled really weird. They didn't grip the road properly and the front end would seem to slide on the pavement.
In another case, a single tire in a set of 10 year old tires with "like new" tread split apart while driving at 30 mph. The tread separated and beat the inner fender of the vehicle to death.