Your problem could be that someone replaced the original rubber with a newer tire with a higher psi rating.

I believe the Phantom did not have a hooked rim wheel which means a tire in some situations is more likely to slip off the rim than modern wheels. With some Schwinn bikes like the Varsity you can fit 90 psi tires but should never exceed 70 or you will most likely have problems.

Never exceed the recommended pressure for original tires and maybe go a little less to allow for pressure gauge inaccuracy.

Also, as stated previously, use wire bead tires.

Of course maybe your tires are very old and if deformed from not being aired up or have a broken bead.