Vibration harvesters typically are linear mass-spring devices working at resonance. A different approach is here proposed based on nonlinear converters that exploit stochastic resonance with white-noise excitation. It consists of a piezoelectric beam converter coupled to permanent magnets to create a bistable system. Under proper conditions, the system bounces between two stable states in response to random excitation, which significantly improves energy harvesting from wide-spectrum vibrations. The background theory is discussed based on a simplified monodimensional model which includes nonlinearity. A cantilever beam with added nonlinearity was simulated by using a MATLAB® Stochastic Differential Equation (SDE) Toolbox demonstrating the expected improvement under white-noise vibrations. Nonlinear converters were then realized by screen printing low-curing-temperature lead zirconate titanate (PZT) films on steel cantilevers equipped with magnets. Experimental tests were performed by measuring both the beam deflection and the output voltage under excitation by random vibrations at varying degree of nonlinearity added to the system. The obtained results show that the performances of the converter in terms of output voltage at parity of mechanical excitation are markedly improved when the system is made bistable. Furthermore, the principle was also preliminarily validated on aMEMSU-shaped cantilever beam that was purposely designed and fabricated in SOI technology. This demonstrates the possibility to downscale the principle here proposed in the perspective of a MEMS harvester based on nonlinear piezoelectric converters.