We present the first calculations of finite length carbon nanotube thermal conductivity that extend from the ballistic to the diffusive regime, throughout a very wide range of lengths and temperatures. The long standing problem of vanishing scattering of the “long wavelength phonons” (Pomeranchuk, I. J. Phys. (U.S.S. R.) 1941, 4, 259; Phys. Rev. 1941, 60, 820) manifests itself dramatically here, making the thermal conductivity diverge as the nanotube length increases. We show that the divergence disappears if 3-phonon scattering processes are considered to second or higher order. Nevertheless, for defect free nanotubes, the thermal conductivity keeps increasing up to very large lengths (10 μm at 300 K). Defects in the nanotube are also able to remove the long wavelength divergence.