By means of nanoindentation, the sudden advances of the tip ("pop-ins") were systematically studied in thin hard films on soft substrates (amorphous aluminum oxide films on aluminum). The critical load and critical indentation depth at the breakthrough of the film were measured as a function of the film thickness. A model, based on the load-induced expansion of the plastic zone in the film, is developed to describe the critical parameters. The model predictions are in good agreement with the measured values for large film thicknesses, while deviations are observed for the thinnest films. These deviations are attributed to a membrane effect in the hard film/soft substrate system. The contribution of this effect is analyzed in comparison with estimates based on a model of a yielding substrate under film deflection and on another model of a "plate on elastic foundation".