The study of dynamical tunneling in a periodically driven anharmonic potential probes the quantum-classical transition via the experimental control of the effective Planck's constant for the system. In this paper we consider the prospects for observing dynamical tunneling with ultracold atoms in magnetic microtraps on atom chips. We outline the driven anharmonic potentials that are possible using standard magnetic traps and find the Floquet spectrum for one of these as a function of the potential strength, modulation, and effective Planck's constant. We develop an integrable approximation to the nonintegrable Hamiltonian and find that it can explain the behavior of the tunneling rate as a function of the effective Planck's constant in the regular region of parameter space. In the chaotic region we compare our results with the predictions of models that describe chaos-assisted tunneling. Finally, we examine the practicality of performing these experiments in the laboratory with Bose-Einstein condensates.