A well-known strategy to improve the electrical conductivity of polymers is to dope them with high-aspect-ratio and conductive nanoparticles such as carbon nanotubes (CNTs). However, these nanocomposites also exhibit undesirable properties such as damage-sensitive and history-dependent conductivity because their macroscopic electrical conductivity is largely determined by the tunneling effect at the tube/tube interface. To reduce these issues, new nanocomposites have been developed with CNTs that have been coated with a conductive layer of poly(3,4-ethylenedioxythiophene)poly(styrenesulfonate) (PEDOT/PSS). It has been posited that the insulating region between the CNTs is replaced by a conductive polymer bridge; this has not been proven up to now. We propose here to investigate in-depth how the macroscopic conductivity of these materials is changing when (1) varying the frequency of the electrical loading (impedance spectroscopy), (2) varying the mechanical hydrostatic pressure, and (3) varying the voltage of the electrical loading. The response is systematically compared to the one of conventional carbon nanotube/polycarbonate (CNT/PC) nanocomposites so we can clarify how efficiently the tunneling effect is suppressed from these composites. The objective is to elucidate further the mechanism for conduction in such material formulations.