Copper/carbon nanofibre composites containing titanium varying from 0.3wt% to 5wt% were made, and their thermal conductivities measured using the laser flash technique. The measured thermal conductivities were much lower than predicted. The difference between measured and predicted values has often been attributed to limited heat flow across the interface. A study has been made of the composite microstructure using x-ray diffraction, transmission electron microscopy and Raman spectroscopy. It is shown in these materials, that the low composite thermal conductivity arises primarily because the highly graphitic carbon nanofibre structure transforming to amorphous carbon during the fabrication process.