I had this happen to me on my old Motobecane once. It had 27" rims too. I had bought a new tire and put it on the rear wheel, had like three blowouts like you describe. I tried and tried to correct the problem and finally gave up and took the new tire back to the bike shop with all the blown out tubes. I told them how my old tire never did this and how the new one will not stay on the rim and showed him the three tubes with star shaped holes in them. (The tire was a wire bead, not a folder) He gave me a new tire (I think it was another kind, can't recall) and three new tubes no charge. It was fine with the new tire he gave me. I've always thought that this was just a case of poor workmanship from the factory. Its never happened to me once since.

Consider replacing the tire. It may just be made badly.