A detailed study on the mechanism of band-to-band tunneling in carbon nanotube field-effect transistors (CNFETs) is presented. Through a dual-gated CNFET structure tunneling currents from the valence into the conduction band and vice versa can be enabled or disabled by changing the gate potential. Different from a conventional device where the Fermi distribution ultimately limits the gate voltage range for switching the device on or off, current flow is controlled here by the valence and conduction band edges in a bandpass-filter-like arrangement. We discuss how the structure of the nanotube is the key enabler of this particular one-dimensional tunneling effect.