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. Because all prior telescopes sensitive at E > 10 keV do not focus light and have degree-scale fields of view, their backgrounds are both high and difficult to characterize. The associated uncertainties result in lower sensitivity to IC emission and a greater chance of false detection. 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. Applying the background model to the Bullet cluster data, we find that the spectrum is well-but not perfectly-described as an isothermal plasma with kT = 14.2 ± 0.2 keV. To slightly improve the fit, a second temperature component is added, which appears to account for lower temperature emission from the cool core, pushing the primary component to kT ~ 15.3 keV. We see no convincing need to invoke an IC component to describe the spectrum of the Bullet cluster, and instead argue that it is dominated at all energies by emission from purely thermal gas. The conservatively derived 90% upper limit on the IC flux of 1.1 × 10-12 erg s-1 cm-2 (50-100 keV), implying a lower limit on B ≳ 0.2 μG, is barely consistent with detected fluxes previously reported. 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.