Copper is an important material for the nuclear industry; therefore, the correct copper cross section are essential. This paper deals with benchmarking the fast neutron leakage spectrum from a copper block with an intense 252Cf source. The spectrum was measured by the proton recoil method using a stilbene scintillator and spectrometer with pulse shape discrimination. The room effect was subtracted experimentally by shielding cone and confirmed computationally. Simulations were performed using the MCNP6.2 Monte Carlo code. A detailed study on uncertainties has been made. Consequently, as the main source of uncertainties, the measurement apparatus was determined (namely, calibration and number of iterations—up to 15%). From the parameter uncertainty, the most important effects are connected with the mass of cube material—the thickness of the assembly (up to 2%). The effect of copper density uncertainty is low because density was determined experimentally with low uncertainty. The uncertainty related to parameter uncertainties in source structural components is negligible because even neglection of all components leads to a shift in leakage spectra below 7%. Several copper nuclear data libraries were tested as well, and it was found that the calculation with JEFF-3.3 gives the most discrepant results with a discrepancy of up to 60%. The closest results were obtained with ENDF/B-VIII.0 (below 5 MeV, discrepancy within 10%) and JENDL-4.0 (above 5 MeV, discrepancy within 15%). ENDF/B-VII.1 library is relatively close in the region above 5 MeV (discrepancy within 15%), below 5 MeV the agreement is worse (up to 40% discrepancy).