We demonstrate that a quiet state and large-amplitude self-sustained oscillations can coexist in a carbon nanotube subject to time-independent drive. A feature of the bistability is that it would be hysteresis free in the absence of noise, and the oscillatory state would not be seen. It is revealed by random switching between the stable states, which we observe in the time domain. We attribute the switching to fluctuations in the system and show that it displays Poisson statistics. We propose a minimalistic model that relates the emergence of the bistability to a nonmonotonic variation of nonlinear friction with the vibration amplitude. This new type of dynamical regime and the means to reveal it are generic and are of interest for various mesoscopic vibrational systems.