The reliability of the mean-field approach to polymer statistical mechanics is investigated by comparing results from a recently developed lattice mean-field theory (LMFT) method to statistically exact results from two independent numerical Monte Carlo simulations for the problems of a polymer chain moving in a spherical cavity and a polymer chain partitioning between two confining spheres of different radii. It is shown that in some cases the agreement between the LMFT and the simulation results is excellent, while in others, such as the case of strongly fluctuating monomer repulsion fields, the LMFT results agree with the simulations only qualitatively. Various approximations of the LMFT method are systematically estimated, and the quantitative discrepancy between the two sets of results is explained with the diminished accuracy of the saddle-point approximation, implicit in the mean-field method, in the case of strongly fluctuating fields.