Hillrider is right. I owned a 1969 Schwinn Super Sport with old style rims, the non-hook bead rim kind. Try to put over the recommended maximum 70 psi and POW - off goes the tire and the tube would be ripped with a huge hole.

I did, however, have the same thing happen on a hook bead rim. It was a Sun rim and no brand of tire would stay on it. It turned out that the rim was made too small, a manufacturing defect. The way Sun handled the problem was very disappointing, but that's another story....

It's easy to check for this type of thing. Measure the perimeter of a known good rim with a sewing type tape (they will lay flat on the OD of the rim, unlike a carpenter type tape measure) and compare to the rim in question.