We have investigated the electrical transport properties of carbon nanotube field-effect transistors as a function of channel length, gate dielectric film thickness, and dielectric material. Our experiments show that the bulk properties of the semiconducting carbon nanotubes do not limit the current flow through the metal/nanotube/metal system. Instead, our results can be understood in the framework of gate and source-drain field induced modulation of the nanotube band structure at the source contact. The existence of one-dimensional Schottky barriers at the metal/nanotube interface determines the device performance and results in an unexpected scaling behavior.