Nanostructured composites containing aligned carbon nanotubes (CNTs) are very promising as interface materials for electronic systems and thermoelectric power generators. We report the first data for the thermal conductivity of densified, aligned multiwall CNT nanocomposite films for a range of CNT volume fractions. A 1 vol % CNT composite more than doubles the thermal conductivity of the base polymer. Denser arrays (17 vol % CNTs) enhance the thermal conductivity by as much as a factor of 18 and there is a nonlinear trend with CNT volume fraction. This article discusses the impact of CNT density on thermal conduction considering boundary resistances, increased defect concentrations, and the possibility of suppressed phonon modes in the CNTs.