We report conductance measurements on multiwall carbon nanotubes in a perpendicular magnetic field. A gate electrode with large capacitance is used to considerably vary the nanotube Fermi level. This enables us to search for signatures of the unique electronic band structure of the nanotubes in the regime of diffusive quantum transport. We find an unusual quenching of the magnetoconductance and the zero bias anomaly in the differential conductance at certain gate voltages, which can be linked to the onset of quasi-one-dimensional subbands.