The cytoskeleton is a highly interconnected meshwork of strongly coupled subsystems providing mechanical stability as well as dynamic functions to cells. To elucidate the underlying biophysical principles, it is central to investigate not only one distinct functional subsystem but rather their interplay as composite biopolymeric structures. Two of the key cytoskeletal elements are actin and vimentin filaments. Here, we show that composite networks reconstituted from actin and vimentin can be described by a superposition of two non-interacting scaffolds. Arising effects are demonstrated in a scale-spanning frame connecting single filament dynamics to macro-rheological network properties. The acquired results of the linear and non-linear bulk mechanics can be captured within an inelastic glassy wormlike chain model. In contrast to previous studies, we find no emergent effects in these composite networks. Thus, our study paves the way to predict the mechanics of the cytoskeleton based on the properties of its single structural components.