Composite material systems composed of a matrix of nanomaterials can achieve combinations of mechanical and thermophysical properties outside the range of traditional systems. The microstructure of the system dictates the rate, in which heat moves through the material. In this work, air/carbon nanofiber networks are studied to elucidate the system parameters influencing thermal transport. Thermal properties are measured with varying initial carbon fiber fill fraction, environment pressure, loading pressure, and heat treatment temperature (HTT) through a bidirectional modification of the 3ω technique. The nanostructure of the individual fibers is characterized with small angle X-ray scattering and Raman spectroscopy providing insight to individual fiber thermal conductivity. Measured thermal conductivity of the carbon nanofiber networks varied from 0.010 W/(m K) to 0.070 W/(m K). An understanding of the intrinsic properties of the individual fibers and the interactions of the two-phase composite is used to reconcile low measured thermal conductivities with predictive modeling. Accounting for fiber-to-fiber interactions and the nuanced changes in the composite as pressure is applied is necessary to successfully model thermal transport in system.