The search for diffuse non-thermal inverse Compton (IC) emission from galaxy clusters at hard X-ray energies has been undertaken with many instruments, with most detections being either of low significance or controversial. In this work, we present 266 ks NuSTAR observations of the Bullet cluster, which is detected in the energy range 3-30 keV. NuSTAR's unprecedented hard X-ray focusing capability largely eliminates confusion between diffuse IC and point sources; however, at the highest energies, the background still dominates and must be well understood. To this end, we have developed a complete background model constructed of physically inspired components constrained by extragalactic survey field observations, the specific parameters of which are derived locally from data in non-source regions of target observations. In addition to discussing the possible origin of this discrepancy, we remark on the potential implications of this analysis for the prospects for detecting IC in galaxy clusters in the future.