The cytoskeleton, an intricate network of protein filaments, motor proteins, and cross-linkers, largely determines the mechanical properties of cells. Among the three filamentous components, F-actin, microtubules, and intermediate filaments (IFs), the IF network is by far the most extensible and resilient to stress. We present a multiscale approach to disentangle the three main contributions to vimentin IF network mechanics-single-filament mechanics, filament length, and interactions between filaments-including their temporal evolution. Combining particle tracking, quadruple optical trapping, and computational modeling, we derive quantitative information on the strength and kinetics of filament interactions. Specifically, we find that hydrophobic contributions to network mechanics enter mostly via filament-elongation kinetics, whereas electrostatics have a direct influence on filament-filament interactions.