Solvent discreteness or "graininess" is usually considered to affect solvation energetics by modifying the intermolecular structure of the solvent, which in turn modifies its dielectric constant and the solute-solvent configurations. In this work, we separate the effect of solvent discreteness from solvent structure and polarity, as well as the arrangement of solvent particles around the solute. Because it is rather diffficult to do this separation with real solutions and their "realistic" models, we utilized translationally fixed dipole lattices, which allow such a separation. The polarity and the dielectric constant of a dipole lattice can be kept invariant as the number density of the dipoles is varied. The lattice spacing represents the degree of discreteness or coarseness in a lattice. Dipole lattices that polarize according to the continuum prediction, as well as truly interacting dipole lattices, lead to effective cavity sizes that are significantly smaller than the geometrically defined exclusion radius (the radius of a sphere around the center of a solute ion into which solvent particles cannot penetrate). The results are similar for lattices of opposite microscopic polarization tendencies and opposite ferroelectric divergence properties. Placing the solute in an interstitial or substitutional position does not cause a qualitative change in the results; increased solvent discreteness leads to smaller Born radii. To check whether this interesting result is peculiar to dipole lattice representations, we studied various forms of solute-solvent distribution function g(r). We derived a formula that connects g(r) to an effective cavity radius, with the approximation that the microscopic polarization follows the continuum prediction. The effective cavity radius is again found to be smaller than the exclusion radius. In agreement with the dipole-lattice analysis, the effective cavity radius decreases with increasing "graininess" of the solute-solvent pair distribution function; wavier g(r)＇s lead to smaller effective cavity sizes. This result has important conceptual and practical implications in solvation modeling. Solvent discreteness becomes an important factor in solvation in its own right, distinct from its indirect effects felt through modified solvent properties and solute-solvent configurations. Also, in light of the present results, it should be possible to develop better parameterization schemes for simplified solvation models.