An attempt is made to understand the underscreening effect, observed in concentrated electrolyte solutions or melts, on the basis of simple, admittedly crude models involving charged (for the ions) and neutral (for the solvent molecules) hard spheres. The thermodynamic and structural properties of these "primitive" and "semi-primitive" models are calculated within the mean spherical approximation (MSA), which provides the basic input required to determine the partial density response functions. The screening length λ S , which is unambiguously defined in terms of the wave-number-dependent response functions, exhibits a cross-over from a low density, Debye-like regime, to a regime where λ S increases with density beyond a critical density at which the Debye length λ D becomes comparable to the ion diameter. In this high density regime the ratio λ S /λ D increases according to a power law, in qualitative agreement with experimental measurements, albeit at a much slower rate.