We report a determination of the irreversibility line in the field-temperature plane of YBa2Cu3O7-x films measured by ac susceptibility. For films thicker than 1000, Hirr is similar to that observed in single crystals even though the critical current in the films is orders of magnitude larger. However, for films thinner than 1000, the irreversibility line is observed to shift to lower temperatures. This suppression is discussed in terms of various models for the irreversibility line. © 1991 The American Physical Society.