Cellulose is a crystalline polymer with intriguing, amorphous-like, temperature dependence of thermal conductivity κ. To determine its origin, we have studied κ of cellulose nanocrystals (CNCs) derived from cotton by sulfuric acid hydrolysis, in both porous and nonporous states by pressure densification; κ increases weakly with increasing temperature and density, like in a fully amorphous material, and it is remarkably similar to that of cellulose fibers (CFs) and cellulose nanofibers (CNFs). For a powder derived from a natural material, like cellulose, amorphous-like κ may originate from poor thermal contact between particles or a high amorphous content, but the latter is not the case for CNCs. Moreover, the amorphous-like behavior is unaffected by densification and, therefore, improved thermal contacts. Instead, we attribute the behavior to CNCs' nanometer-sized fibrils, which limit the phonon mean free path to a few nanometers in a network of randomly oriented CNCs. This explains why κ is essentially the same in networks of CNCs, CFs, and CNFs, which are materials with the same structural unit-elementary fibrils of 3-5 nm in diameter. We obtain κ = (0.60 ± 0.01) W m-1 K-1 for a nonporous network of randomly oriented CNCs at 295 K and atmospheric pressure, and κ increases by only 14% GPa-1, which is unusually weak for a polymer. By using a model for such a network, we find κ = 1.9 W m-1 K-1 along a CNC and argue that this is a good estimate also along a CNF and a CF at room temperature.