The modified Poisson-Boltzmann theory is applied to a study of the cell model of linear polyelectrolyte solutions. The theoretical results for polyion-simple ion correlations are compared with those obtained in parallel Monte Carlo simulations and classical Poisson-Boltzmann calculations for the same model. The modified Poisson-Boltzmann predictions are found to be in complete agreement with the simulation data for monovalent simple ions and moderate concentrations, while the Poisson-Boltzmann theory is valid at low concentrations. In solutions containing divalent ions, where standard theories tend to give a very rough estimate of the simulated structure, the modified Poisson-Boltzmann results remain very close to those obtained from the simulations. The theory appears to provide an essentially correct account of interionic correlations, which are neglected in mean field theory, such as the Poisson-Boltzmann approximation, leading to poor results. The calculations for high ionic strengths illustrate the limitations of the cell model due to the neglect of the correlations between double layers surrounding neighboring polyions in solution.