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Carbon nanotubes (CNTs) are quasi-one dimensional nanostructures that display both high thermal conductivity for potential thermal management applications and intriguing low-dimensional phonon transport phenomena. In comparison to the advances made in the theoretical calculation of the lattice thermal conductivity of CNTs, thermal transport measurements of CNTs have been limited by either the poor temperature sensitivity of Raman thermometry technique or the presence of contact thermal resistance errors in sensitive two-probe resistance thermometry measurements. Here we report advances in a multi-probe measurement of the intrinsic thermal conductivity of individual multi-walled CNT samples that are transferred from the growth substrate onto the measurement device. The sample-thermometer thermal interface resistance is directly measured by this multi-probe method and used to model the temperature distribution along the contacted sample segment. The detailed temperature profile helps to eliminate the contact thermal resistance error in the obtained thermal conductivity of the suspended sample segment. A differential electro-thermal bridge measurement method is established to enhance the signal-to-noise ratio and reduce the measurement uncertainty by over 40%. The obtained thermal resistances of multiple suspended segments of the same MWCNT samples increase nearly linearly with increasing length, revealing diffusive phonon transport as a result of phonon-defect scattering in these MWCNT samples. The measured thermal conductivity increases with temperature and reaches up to 390 ± 20 W m-1 K-1 at room temperature for a 9-walled MWCNT. Theoretical analysis of the measurement results suggests submicron phonon mean free paths due to extrinsic phonon scattering by extended defects such as grain boundaries. The obtained thermal conductivity is decreased by a factor of 3 upon electron beam damage and surface contamination of the CNT sample.