This paper describes the results of calibration measurements for a Bonner sphere spectrometer (BSS) with 3He proportional counter performed in quasi-monoenergetic neutron fields at the Research Center for Nuclear Physics (RCNP) at the University of Osaka, Japan. Using 246 MeV and 389 MeV proton beams, neutron fields with nominal peak energies of 244 MeV and 387 MeV were generated via 7Li(p,n)7Be reactions. At high energies, the neutron spectra were measured by means of the time-of-flight (TOF) method. The low-energy part of the neutron spectra were determined by BSS measurements down to thermal energies using the MSANDB unfolding code and three different sets of response functions. These were obtained by means of Monte Carlo (MC) calculations including various codes and intra-nuclear cascade (INC) models. Unfolded BSS fluence rates were additionally confirmed by GEANT4 calculations. For calibration of the BSS, measured count rates were corrected for low-energy contributions and compared with count rates calculated using TOF data and various response functions. In addition, measured response values were compared with mono-energetic response calculations, and best agreement was found with GEANT4 results using the Bertini INC model.