Computational study of thermal conductivity of interconnected networks of bundles in carbon nanotube (CNT) films reveals a strong effect of the finite thermal conductivity k(T) of individual nanotubes on the conductivity k of the CNT materials. The physical origin of this effect is explained in a theoretical analysis of systems composed of straight randomly dispersed CNTs. An analytical equation for quantitative description of the effect of finite k(T) on the value of k is obtained and adopted for continuous networks of bundles characteristic of CNT films and buckypaper. Contrary to the common assumption of the dominant effect of the contact conductance, the contribution of the finite k(T) is found to control the value of k at material densities and CNT lengths typical for real materials. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4737903]