Broadband X-ray spectroscopy of the X-ray emission produced in the coronae of active galactic nuclei (AGNs) can provide important insights into the physical conditions very close to their central supermassive black holes. The temperature of the Comptonizing plasma that forms the corona is manifested through a high-energy cutoff that has been difficult to directly constrain even in the brightest AGN because it requires high-quality data at energies above 10 keV. In this paper we present a large collection of coronal cutoff constraints for obscured AGNs based on a sample of 130 AGNs selected in the hard X-ray band with Swift/BAT and observed nearly simultaneously with NuSTAR and Swift/XRT. We find that under a reasonable set of assumptions regarding partial constraints the median cutoff is well constrained to 290 20 keV, where the uncertainty is statistical and given at the 68% confidence level. We investigate the sensitivity of this result to our assumptions and find that consideration of various known systematic uncertainties robustly places the median cutoff between 240 and 340 keV. The central 68% of the intrinsic cutoff distribution is found to be between about 140 and 500 keV, with estimated uncertainties of 20 and 100 keV, respectively. In comparison with the literature, we find no clear evidence that the cutoffs in obscured and unobscured AGNs are substantially different. Our analysis highlights the importance of carefully considering partial and potentially degenerate constraints on the coronal high-energy cutoff in AGNs.