We study a version of the Colonel Blotto game where valuations across battlefields are heterogeneous and asymmetric. These games can exhibit unique pure strategy equilibria, some of which are non-monotonic with respect to the battlefield valuations. We test our theoretical predictions in the laboratory and find low initial levels of equilibrium play but substantial learning throughout the experiment. Learning is higher for games with monotonic equilibria. Finally, we find that deviations from equilibrium predictions benefit aggregate welfare.