The extremely high thermal conductivity of individual carbon nanotubes, predicted theoretically and observed experimentally, has not yet been achieved for large nanotube assemblies. Resistances at tube-tube interconnections and tube-electrode interfaces have been considered the main obstacles for effective electronic and heat transport. Here we show that, even for infinitely long and perfect nanotubes with well-designed tube-electrode interfaces, excessive radial heat radiation from nanotube surfaces and quenching of phonon modes in large bundles are additional processes that substantially reduce thermal transport along nanotubes. Equivalent circuit simulations and an experimental self-heating 3omega technique were used to determine the peculiarities of anisotropic heat flow and thermal conductivity of single MWNTs, bundled MWNTs and aligned, free-standing MWNT sheets. The thermal conductivity of individual MWNTs grown by chemical vapor deposition and normalized to the density of graphite is much lower (kappa(MWNT) = 600 +/- 100 W m(-1) K(-1)) than theoretically predicted. Coupling within MWNT bundles decreases this thermal conductivity to 150 W m(-1) K(-1). Further decrease of the effective thermal conductivity in MWNT sheets to 50 W m(-1) K(-1) comes from tube-tube interconnections and sheet imperfections like dangling fiber ends, loops and misalignment of nanotubes. Optimal structures for enhancing thermal conductivity are discussed.