Bike Help Needed

From a story that begins well with the Bikerowave, and ends today:

Less awesome: The bike I was working on. I brought in an absolute clunker of a bike – an old Motobecane that had been sitting outside under the porch in Colorado for far too long and had then made the trip to California and spent most of its time sitting in our hallway. Less than awesome. There’s progress on the beast, and it was almost (briefly) rideable. But when I brought it home and inflated the tire a little more, I realized there was a slight issue. If the tire was inflated to the recommended psi (say, about 80), it bulged out seriously on the sidewall, so much so that it wouldn’t spin through the brakes. So I deflated the tire, tried to reseat the bead in the rim, then inflated the tire again. Same problem. Repeat, except this time I removed the tire and spread a little chalk on the inside between the tube and the tire, thinking that the tube might be getting bunched up when inflated. When I inflated the tire again (again, to about 80 psi, though the tire says it takes 90 psi), it seemed fine. I was psyched. Deflate the tire, slipped it back into the bike, tightened down the wheel a little, then pumped up the tire again. Everything seemed gravy, but within about a minute, the tire/tube was bulging back over the rim. I went to deflate the tire again to look at it when the tube (predictably?) popped.

Any thoughts?