We investigate experimentally the adherence energy Gamma of model polyacrylate pressure-sensitive adhesives (PSAs) with combined large strain rheological measurements in uniaxial extension and an instrumented peel test. We develop a nonlinear model for such a peel test which captures the dependence of Gamma(V) with peeling rate V, revealing the key role played by the extensional rheology. Our model explains in particular why traditional linear viscoelastic approaches correctly predict the slope of Gamma(V) curves for sufficiently elastic PSAs characterized by a simple rate-independent debonding criterion. However, for more viscoelastic adhesives, we identified a more complex rate-dependent debonding criterion yielding a significant modification of the Gamma(V) curves, an effect that has been largely overlooked so far. This investigation opens the way toward the understanding of fibrils debonding, which is the main missing block to predict the adherence of PSAs.