A suspended, doubly clamped single-wall carbon nanotube is characterized at cryogenic temperatures. We observe specific switching effects in dc-current spectroscopy of the embedded quantum dot. These have been identified previously as nano-electromechanical self-excitation of the system, where positive feedback from single-electron tunneling drives mechanical motion. A magnetic field suppresses this effect, by providing an additional damping mechanism. This is modeled by eddy current damping, and confirmed by measuring the resonance quality factor of the radio-frequency-driven nano-electromechanical resonator in an increasing magnetic field.