An inflatable cushioning article is made by a process of extruding two multilayer films (or extruding one film which is either annular or folded over) each having (a) a seal layer, (b) a tie layer containing an anhydride modified olefin polymer containing anhydride at a level of at least 150 ppm, based on the weight of the modified olefin polymer, and (c) an oxygen barrier layer comprising crystalline polyamide, crystalline polyester, ethylene/vinyl alcohol copolymer, polyacrylonitrile, and/or crystalline polycycloolefin. Selected portions of the films are heat sealed to one another in a selected area providing a heat seal pattern which leaves inflatable chambers between the films, whereby an inflatable cellular cushioning article is produced. At some point after extrusion, at least one of the multilayer films are aged for a time and at a temperature in accordance with at least one member selected from the group consisting of: (i) 141 F. to 250 F. for a period of at least 1 second; (ii) 101 F. to 140 F. for a period of at least 10 minutes; (iii) 61 F. to 100 F. for a period of at least 1 hour; and (iv) 30 F. to 60 F. for a period of at least 1 day. After aging, the cellular cushioning article is inflated. Preferably, the article is inflated to an internal pressure of at least 1.5 psi.