The transport of ultra-cold atoms in magneto-optical potentials provides a clean setting in which to investigate the distinct predictions of classical versus quantum dynamics for a system with coupled degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from the coupling between the atomic spin and its center-of-mass motion. Experiments, performed deep in the quantum regime, correspond to dynamic quantum tunneling. This nonclassical behavior is contrasted with the predictions for an initial phase space distribution produced in the experiment, but undergoing classical Hamiltonian flow. We study conditions under which the trapped atoms can be made to exhibit classical dynamics through the process of continuous measurement, which localizes the probability distribution to phase space trajectories, consistent with the uncertainty principle and quantum "back-action" noise. This method allows us to analytically and numerically identify the quantum-classical boundary.