When you say "TEST DRIVE" do you mean that this was a ... P.O.S. bike you had found in the trash? ... Bike you had not before ridden? ... New purchase? ... New wheel? ... New tire? ... New tube?

One first considerstion: There were TWO different and completely incompatible rims and tires which were BOTH called 26 x 1-3/8"

1.) Raleighs and most British 3-speed bikes used an "EA.3" rim for a 26 x 1-3/8" tire which had a bead diameter of 590. [ISO: 37x590]

2.) Schwinn 3-speeds bikes used their "S-6" size rim for their totally different size tire they ALSO called a 26 x 1-3/8" which had a bead diameter of 597 [ISO: 37x597]

Put the minutely larger size tire on the smaller rim, inflate it well, and it might ride briefly until one little section of tire slips out of the rim, and BAM! I've had this happen even with the correct sized modern road tires which simply had not seated quite right into a perfectly correct rim. To compound matters, those older rims often did not have nice things like modern "hook edged" rims which help to hold the bead of the tire from slipping off. If the tire were old and neglected, a section of the sidewalls could have weakened to allow the tire to bulge out at one spot, and the bead itself could even have separated or loosened from the sidewall. I suppose a ding on the edge where the tire and rim meet might give enough variance to release a section of tire too. And then over-inflating a tire could possibly push apart the sides enough to help the tire jump the rim.

I would carefully check that the tire and rim are a suitable match. Then, carefully examine the tire for any wear or damage - especially after a blow-out. Then, after replacing the tube, inflate it slowly, in stages, and keep checking that it is staying on the rim on both sides and all around. ... Then pray.