We theoretically predict the nonlinear elastic responses of polydisperse biopolymer gels to uniaxial compression. We analyze the competition between compressive stiffening due to polymer densification by out-going solvent flow and compressive softening due to continuous polymer buckling. We point out that the polydispersity in polymer lengths can result in an intrinsic, equilibrium mode of nonaffine compression: nonuniform strain but with uniform force distribution, which is found to be more energetically preferable than affine deformation. In this case, the gel softens significantly after the onset of polymer buckling at small compression, but as compression increases, densification-induced stiffening becomes important and a modulus plateau should be observed for a large range of strain. We also relate our results to recent compression experiments on collagen gels and fibrin gels.