We show that rapid decoherence, usually associated with chaotic dynamics, is not necessarily a hallmark of nonintegrability: border effects in integrable systems may produce similarly drastic decoherence rates. These can be found when the subsystem under observation possesses an energy limitation as, e.g., in the N-atom Jaynes-Cummings model. We show for this model that special initial coherent wave packets exhibit entropy production rates strikingly similar to the chaotic case. Also, a (de)localization phenomenon is found to be a function of the proximity to the phase-space border. [S1063-651X(99)01611-6].605A54075411