Large enchancements in certain elastic moduli (by more than 100%) of compositionally modulated films of certain metal alloys, in particular copper-nickel, have been reported. Among the explanations proposed for the phenomenon, it has been suggested that the enhancement is due to coherency-induced strains and preliminary calculations based on nonlinear continuum elasticity have been reported in support of this idea. We have explored this mechanism in detail for copper-nickel superlattices. The variation of the elastic constants with strain (including up to fourth-order elastic constants) as well as with alloy composition have been considered. The average elastic moduli have been determined for a sinusoidally modulated film. We have found that the biaxial moduli Y[001] and Y[111] vary as the square of the modulation amplitude (A). The biaxial modulus Y[001] shows a small enhancement, on the order of a few percent, while Y[111] actually decreases, by a few percent, from its value for the homogeneous alloy. We have also found a similar decrease of Y[111] for layered copper-nickel superlattices with sharp interfaces. The relative contributions of the strain and composition dependences of the elastic constants to the net change have been investigated. While the strain dependence alone increases the modulus of the modulated film, the composition dependence decreases it. The third- and fourth-order elastic constants (reflecting the strain dependence of the elastic moduli) have to be much larger than their experimentally determined values (even accounting for experimental uncertainty) in order to get even a small enhancement of Y[111]. We conclude that the enhancement cannot be explained by a continuum model of the coherency-strained superlattice, but note that an atomic-scale effect cannot be ruled out.