A supply of inflatable pouches comprising: an inflatable web comprising two sheets (20a, 20b) having inner surfaces (22a, 22b) sealed to each other in a pattern defining a series of inflatable chambers, each chamber having a closed distal end and an open proximal end, the open proximal end providing an inflation port for each of the chambers, the chambers arrayed in a transverse orientation to a longitudinal dimension of the web; a longitudinal fold in the web, which extends along the longitudinal dimension thereof such that the web is configured in the form of two juxtaposed inflatable panels (238a, 238b) joined together at the longitudinal fold; and a series of transverse seals that bond the panels together to form a connected series of inflatable pouches between pairs of the transverse seals such that the connected series of inflatable pouches extends along the longitudinal dimension of the web, each inflatable pouch comprising one or more of the inflatable chambers; characterized by a series of slits, each slit extending transversely in the two juxtaposed inflatable panels (238a, 238b) from the longitudinal fold for at least 2.54 centimeters, each slit being between adjacent inflatable pouches; and a series of lines of weakness extending transversely across the web and between adjacent inflatable pouches; wherein, as the inflatable pouches are inflated, the width of the inflatable pouches shortens as the inflation gas expands the chambers; wherein the lines of weakness are situated in a region of the web that does not shrink; and wherein each slit of the series of slits corresponds with a respective line of weakness of the series of lines of weakness. an inflatable web comprising two sheets (20a, 20b) having inner surfaces (22a, 22b) sealed to each other in a pattern defining a series of inflatable chambers, each chamber having a closed distal end and an open proximal end, the open proximal end providing an inflation port for each of the chambers, the chambers arrayed in a transverse orientation to a longitudinal dimension of the web; a longitudinal fold in the web, which extends along the longitudinal dimension thereof such that the web is configured in the form of two juxtaposed inflatable panels (238a, 238b) joined together at the longitudinal fold; and a series of transverse seals that bond the panels together to form a connected series of inflatable pouches between pairs of the transverse seals such that the connected series of inflatable pouches extends along the longitudinal dimension of the web, each inflatable pouch comprising one or more of the inflatable chambers; characterized by a series of slits, each slit extending transversely in the two juxtaposed inflatable panels (238a, 238b) from the longitudinal fold for at least 2.54 centimeters, each slit being between adjacent inflatable pouches; and a series of lines of weakness extending transversely across the web and between adjacent inflatable pouches; wherein, as the inflatable pouches are inflated, the width of the inflatable pouches shortens as the inflation gas expands the chambers; wherein the lines of weakness are situated in a region of the web that does not shrink; and wherein each slit of the series of slits corresponds with a respective line of weakness of the series of lines of weakness.