Quasi-monoenergetic high-energy neutron fields induced by Li-7(p,n) reactions are used for the response evaluation of neutron-sensitive devices. The quasi-monoenergetic high-energy field consists of high-energy monoenergetic peak neutrons and unwanted continuum neutrons down to the low-energy region. A two-angle differential method has been developed to compensate for the effect of the continuum neutrons in the response measurements. In this study, the two-angle differential method was demonstrated for Bonner sphere detectors, which are typical examples of moderator-based neutron-sensitive detectors, to investigate the method's applicability and its dependence on detector characteristics. Experiments were performed under 96-387 MeV quasi-monoenergetic high-energy neutron fields at the Research Center for Nuclear Physics (RCNP), Osaka University. The measurement results for large high-density polyethylene (HDPE) sphere detectors agreed well with Monte Carlo calculations, which verified the adequacy of the two-angle differential method. By contrast, discrepancies were observed in the results for small HDPE sphere detectors and metal induced sphere detectors. The former indicated that detectors that are particularly sensitive to low-energy neutrons may be affected by penetrating neutrons owing to the geometrical features of the RCNP facility. The latter discrepancy could be consistently explained by a problem in the evaluated cross-section data for the metals used in the calculation. Through those discussions, the adequacy of the two-angle differential method was experimentally verified, and practical suggestions were made pertaining to this method.