The authors use time-resolved cavity-enhanced magneto-optical Kerr spectroscopy to study the damping of magnetization precession in individual cylindrical nickel nanomagnets. A wide range of shapes (diameters of 5 mum-125 nm and aspect ratio: 0.03-1.2) is investigated. They observe a pronounced difference in damping between the micro- and nanomagnets. Microscale magnets show large damping at low bias fields, whereas nanomagnets exhibit bias field-independent damping. This behavior is explained by the interaction of in-plane and out-of-plane precession modes in microscale magnets that results in additional dissipative channels. The small and robust damping values on the nanoscale are promising for implementation of controlled precessional switching schemes in nanomagnetic devices.