Continuum theories of electrolytes are widely used to describe physical processes in various biological systems, even when the system dimensions are comparable to the Debye length. We test the validity of the mean field approximation in Poisson-Boltzmann and Poisson-Nernst-Planck theories by contrasting their predictions with those of Brownian dynamics simulations in cylindrical pores of varying radius. We find that both continuum theories largely overestimate shielding effects when the pore radius is smaller than two Debye lengths, and, therefore, they cannot be used to describe the physics of electrolytes in nanopores.