Thin bilayers of contrasting nanomaterials are ubiquitous in solution-processed electronic devices and have potential relevance to a number of applications in flexible electronics. Motivated by recent mesoscopic simulations demonstrating synergistic mechanical interactions between thin films of single-wall carbon nanotubes (SWCNTs) and spherical nanocrystal (NC) inclusions, we use a thin-film wrinkling approach to query the compressive mechanics of hybrid nanotube/nanocrystal coatings adhered to soft polymer substrates. Our results show an almost 2-fold enhancement in the Young modulus of a sufficiently thin SWCNT film associated with the presence of a thin interpenetrating overlayer of semiconductor NCs. Mesoscopic distinct-element method simulations further support the experimental findings by showing that the additional noncovalent interfaces introduced by nanocrystals enhance the modulus of the SWCNT network and hinder network wrinkling.