We have studied the mechanical behavior of multi-layered composite films comprised of single-wall carbon nanotubes and polystyrene polymer, where we focused on three targeted layer thicknesses; 10 nm, 20 nm, and 40 nm. The approach we used is the Strain-Induced Elastic Buckling Instability for Mechanical Measurements (SIEBIFMM) technique, which allows us to measure the Young modulus of the films as a function of layer thickness and the number of layers by inducing a compressive stress in the films at different strains; 1 %, 2.5 %, 5 %, 7.5 %, and 10 %. Polystyrene was added in an effort to reduce the plasticity of the carbon nanotube films by filling the pores of the nanotube network. We found that the strongest synergistic effect in this regard occurred for the composite with a layer thickness near 20 nm, while the composite films have reached bulk behavior by the sixth layer.