Many biomaterials formed by cross-linked semiflexible or rigid filaments exhibit nonlinear theology in the form of strain-stiffening and negative normal stress when samples are deformed in simple shear geometry. Two different classes of theoretical models have been developed to explain this nonlinear elastic response, which is neither predicted by rubber elasticity theory nor observed in elastomers or gels formed by flexible polymers. One model considers the response of isotropic networks of semiflexible polymers that have nonlinear force-elongation relations arising from their thermal fluctuations. The other considers networks of rigid filaments with linear force-elongation relations in which nonlinearity arises from nonaffine deformation and a shift from filament bending to stretching at increasing strains. Fibrin gels are a good experimental system to test these theories because the fibrin monomer assembles under different conditions to form either thermally fluctuating protofibrils with persistence length on the order of the network mesh size, or thicker rigid fibers. Comparison of rheologic and optical measurements shows that strain stiffening and negative normal stress appear at smaller strains than those at which filament orientation is evident from birefringence. Comparisons of shear to normal stresses and the strain-dependence of shear moduli and birefringence suggest methods to evaluate the applicability of different theories of rod-like polymer networks. The strain-dependence of the ratio of normal stress to shear stress is one parameter that distinguishes semiflexible and rigid filament models, and comparisons with experiments reveal conditions under which specific theories may be applicable.