We present experimental results demonstrating that the flux of a continuous atom laser beam cannot be arbitrarily increased by increasing the strength of output coupling. We find that the continuous atom laser has large density fluctuations when the Rabi frequency becomes greater than the radial trapping frequency, due to the output coupling populating all accessible Zeeman states. In addition, we find that a state-changing output coupler for a continuous atom laser switches off as the output-coupling strength, parametrized in general by the Rabi frequency, is increased above a critical value. The numerical solution of a one-dimensional five-state Gross-Pitaevskii model for the atom laser is in good qualitative agreement with the experiment.