That may have been true at one time, but I do not believe it is true any longer, at least not with steel belted radials. Although there is some variation in the details, most tire makers say you should never run tires that are more than 10 years old, no matter how they were stored or how good they look.
Apparently, air does slowly permeate the rubber, and can cause the carcass to degrade, down under the rubber where it cannot be seen. When such tires are pressed into use, the result can be sudden, severe failure of the carcass.
I have experimented with this. When the LR tire on my first Stag flew apart, the steel belts turned into a giant "flap wheel" (like those things sold on television to remove paint), which very effectively removed not only all the paint but much of the metal from the outside of the wheel opening, before I could get pulled over to the side of the road.
Here is another, milder case, with minimal damage to the car: