The energy spectra of neutrons that penetrate 25- to 200-cm-thick concrete shields are measured using 40- and 65-MeV quasi-monoenergetic neutron sources at the 90-MeV AVF cyclotron of the Takasaki Ion Accelerator Facility for Advanced Radiation Application at the Japan Atomic Energy Research Institute. Source neutrons are produced at 3.6- and 5.2-mm-thick 7Li targets bombarded 43- and 68-Me Vprotons, respectively, and their spectra are measured with a proton recoil counter telescope and the time-of-flight method. In the shielding experiment, a BC501A organic liquid scintillator and a multimoderator spectrometer with a 3He counter (the Bonner Ball) are used as the neutron spectrometer. The collimated source neutrons are used to obtain the neutron spectra with the unfolding technique on the neutron beam axis and at off-center positions. MORSE Monte Carlo calculations are performed with the DLC119/HIL086 multigroup cross-section library for comparison with the measured data on the beam axis. The comparison reveals that the calculated spectra are in good agreement with the measured spectra. The measured data at off-center positions are used to check the calculational accuracy of the angular distribution of neutron scattering in the MORSE-CG, DOT3.5 discrete ordinates, and HETC-KFA2 Monte Carlo codes. The MORSE-CG code gives better results than the other two codes.