This paper addresses the uniaxial compressive failure of unidirectional, fiber-reinforced polymer matrix composites. The material that was used in this combined experimental/numerical effort was APC-2/AS4. Failure was found to result in kink bands of distinct widths oriented at an angle to the line of loading. Kink bands were preserved by conducting a series of experiments where failure was confined. Fiber misalignments in the prepreg and from the lamination process were identified as the imperfections that, in conjunction with the nonlinear shear response of the matrix material, essentially govern the formation of zones of localized bending. This, in turn, causes fibers to break and into distinct kink bands. The process was modeled by considering a micro-section of material with wavy fibers under compression. The formulation accounted for material and geometric nonlinearities and various distributions of waviness. It was illustrated that failure is associated with a limit load instability, followed by localized fiber bending which leads to kinking.