Integral shielding experiments of spent-fuel shipping casks were carried out with a californium source. The measurements of dose rates were performed not only with a cask as designed but also with one having lost its resin shield. The measured neutron and secondary gamma-ray dose rates are compared with the results of Monte Carlo calculations using the next-event surface crossing (NESX) estimation and the usual point detector estimation. Overall, the Monte Carlo-NESX calculation method was found to give better results. The calculated neutron doses from the undamaged cask were in close agreement with the measured values; the agreement was also good in the case of the damaged cask in the radial and axial directions. In particular, the agreement was quite satisfactory at distances up to 100 cm from the cask surface, although the calculated dose rates were a little smaller than the measured values at locations beyond the cask. Nevertheless, the values agreed with the measured ones within a factor of 2. Furthermore, the calculated secondary gamma-ray dose rates using NESX corresponded closely to the measured values for the undamaged cask. With the present knowledge of Monte Carlo techniques, the method could be employed as an effective means of analyzing the radiation shielding of a cask. In addition, the present experimental data can be adopted as a benchmark for cask design.