Experiments have shown that charge distributions of granular materials are non-Gaussian, with broad tails that indicate many particles with high charge. This observation has consequences for the behavior of granular materials in many settings, and may bear relevance to the underlying charge transfer mechanism. However, there is the unaddressed possibility that broad tails arise due to experimental uncertainties, as determining the shapes of tails is nontrivial. Here we show that measurement uncertainties can indeed account for most of the tail broadening previously observed. The clue that reveals this is that distributions are sensitive to the electric field at which they are measured; ones measured at low (high) fields have larger (smaller) tails. Accounting for sources of uncertainty, we reproduce this broadening in silico. Finally, we use our results to back out the true charge distribution without broadening, which we find is still non-Guassian, though with substantially different behavior at the tails and indicating significantly fewer highly charged particles. These results have implications in many natural settings where electrostatic interactions, especially among highly charged particles, strongly affect granular behavior.