When I purchased my 1987 Free Spirit Pinnacle at a garage sale, it still had the original "Golden Boy" tires mounted on it. I aired them up and went out for a ride.
It wasn't long before the front tire let go. A rock hit it and popped the original tube. The front tire and both tubes were changed. I actually got quite a few miles on the original rear tire before I replaced it after it was worn down to the nylon cords. Seriously, when I replaced it, probably about 45% of the tire was just the nylon cords. But it still held air, and somehow I didn't pop any tubes.
Oddly enough, it seemed like I had more flats after changing that original rear tire than before.
But I'm a bad example, so I'd say change them unless you stay fairly close to home and don't go very fast. And even then go ahead and order new tires... they're not super expensive.