In the pursuit of advanced polymer composites, nanoscale fillers have long been championed as promising candidates for structural reinforcement. Despite progress, questions remain as to how these diminutive fillers influence the distribution of stresses within the matrix and, in turn, influence bulk mechanical properties. The dynamic mechanical behavior of elastomer-impregnated forests of carbon nanotubes (CNTs) has revealed distinct orientation-dependent behavior that sheds light on these complicated interactions. When compressed along the axis of the fillers, the composite will mimic open-cell foams and exhibit strain softening for increasing amplitudes due to the collective Euler buckling of the slender nanotubes. In contrast, the same material will behave similarly to the neat polymer when compressed orthogonal to the alignment direction of the nanotubes. However, in this orientation the material is incapable of achieving the same ultimate compressive strain due to the role that the embedded nanotubes play in augmenting the effective cross-link density of the polymer network. Both of these responses are recoverable, robust, and show little dependency on the diameter and wall-number of the included CNTs. Such observations give insight into the mechanics of polymer/nanoparticle interactions in nanocomposite structures under strain, and the thoughtful control of such coordinated buckling behavior opens the possibility for the development of foam-like materials with large Poisson ratios.