Extended-range Bonner sphere spectrometers are well suited for spectrometry in neutron fields that extend from thermal energies to a few hundred mega-electron-volts. The data analysis, however, is not straightforward, and it is of interest to evaluate how well the main features of the spectrum can be determined by the measurements and to estimate the uncertainties of integral quantities of interest, e.g., fluence and dose. In this paper, we apply Bayesian parameter estimation to this problem. We use simulated data that model measurements made in neutron fields behind shielding at high-energy accelerators.