Nanoelectromechanical resonators have potential applications in sensing, cooling, and mechanical signal processing. An important parameter in these systems is the strength of coupling the resonator motion to charge transport through the device. We investigated the mechanical oscillations of a suspended single-walled carbon nanotube that also acts as a single-electron transistor. The coupling of the mechanical and the charge degrees of freedom is strikingly strong as well as widely tunable (the associated damping rate is approximately 3 x 10(6) Hz). In particular, the coupling is strong enough to drive the oscillations in the nonlinear regime.