Simple synthetic and natural hydrogels can be formulated to have elastic moduli that match biological tissues, leading to their widespread application as model systems for tissue engineering, medical device development, and drug delivery vehicles. However, two different hydrogels having the same elastic modulus but differing in microstructure or nanostructure can exhibit drastically different mechanical responses, including their poroelasticity, lubricity, and load bearing capabilities. Here, we investigate the mechanical response of collagen-1 networks to local and bulk compressive loads. We compare these results to the behavior of polyacrylamide, a fundamentally different class of hydrogel network consisting of flexible polymer chains. We find that the high bending rigidity of collagen fibers, which suppresses entropic bending fluctuations and osmotic pressure, facilitates the bulk compression of collagen networks under infinitesimal applied stress. These results are fundamentally different from the behavior of flexible polymer networks in which the entropic thermal fluctuations of the polymer chains result in an osmotic pressure that must first be overcome before bulk compression can occur. Furthermore, we observe minimal transverse strain during the axial loading of collagen networks, a behavior reminiscent of open-celled cellular solids. Inspired by these results, we applied mechanical models of cellular solids to predict the elastic moduli of the collagen networks and found agreement with the moduli values measured through contact indentation. Collectively, these results suggest that unlike flexible polymer networks that are often considered incompressible, collagen hydrogels behave like rigid porous solids that volumetrically compress and expel water rather than spreading laterally under applied normal loads.