For carbon nanotube transistors, as for graphene, the electrical contacts are a key factor limiting device performance. We calculate the device characteristics as a function of nanotube diameter and metal work function. Although the on-state current varies continuously, the transfer characteristics reveal a relatively abrupt crossover from Schottky to Ohmic contacts. We find that typical high-performance devices fall surprisingly close to the crossover. Therefore, tunneling plays an important role even in this regime, so that current fails to saturate with gate voltage as was expected due to "source exhaustion." © 2013 American Physical Society.