When loaded in simple shear deformation, polymeric materials may develop so-called normal stresses: stresses perpendicular to the direction of the applied shear. These normal stresses are intrinsically nonlinear: basic symmetry considerations dictate they may only enter at O(γ(2)), with γ the dimensionless shear strain. There is no fundamental restriction on their sign, and normal stresses may be positive (pushing outward) or negative (pulling inward). Most materials tend to dilate in the normal direction, but a wide variety of biopolymer networks including fibrin and actin gels have been reported to present anomalously large, negative normal stresses-a feature which has been ascribed to the intrinsic elastic nonlinearity of semiflexible fibers. In this work, we present analytical results on a model nonlinear network, which we expand to the required nonlinear order to show that due to geometric, rather than elastic, nonlinearities (negative) normal stresses generically arise in filamentous networks-even in networks composed of linear, Hookean springs. We investigate analytically and numerically how the subsequent addition of elastic nonlinearities, nonaffine deformations, and filament persistence through cross-linkers augment this basic behavior.