Low frequency noise in MOSFETs is believed to be dominated by the effect of traps near the silicon/silicon dioxide interface. Nanometer-scale transistors should exhibit very low trap populations, even zero in some cases, and this should significantly affect the noise properties of such transistors. We have built cylindrical surrounding-gate transistors with channel area smaller than the size in which, on average, one trap would be present at typical trap densities. We observed a reduction in noise by two orders of magnitude when a trap is rendered inactive by biasing. In six of seven devices, the measured power spectral density of the drain current is more than an order of magnitude lower than that predicted using typical trap densities. The drain current still shows random telegraph signals and has regions of Lorentzian and 1/f power spectral density, indicating at least five active traps. However, typical trap densities and the known area of the channel lead to a prediction of only 0.06 active traps. Therefore all six devices must be populated by traps which have much smaller effect on drain current than predicted.