Measurements of thermal transport through contacts between individual multiwall carbon nanotubes show that, contrary to common expectation, the normalized contact thermal conductance per unit area depends linearly on the tube diameter. The result is corroborated with and extended to multilayer graphene nanoribbons through molecular dynamics simulations. Semiquantitative analyses show that these intriguing observations are consistent with an explanation based on an unexpectedly large phonon mean free path in the $c$-axis direction of graphite, phonon reflection at free surfaces, and phonon focusing in highly anisotropic graphitic materials.