The nonlinear elasticity of thin supported membranes assembled from length purified single-wall carbon nanotubes is analyzed through the wrinkling instability that develops under uniaxial compression. In contrast with thin polymer films, pristine nanotube membranes exhibit strong softening under finite strain associated with bond slip and network fracture. We model the response as a shift in percolation threshold generated by strain-induced nanotube alignment in accordance with theoretical predictions.