This work considers the performance of a resonant vibration-based energy harvesting system utilizing a strongly nonlinear attachment. Typical designs serving as the basis for harvesting energy from ambient vibration typically employ a linear oscillator for this purpose, limiting peak harvesting performance to a narrow band of frequencies about the resonant frequency of the oscillator. Herein, in an effort to maximize performance over the broader band of frequency content typically observed in ambient vibration measurements, we employ an essentially nonlinear cubic oscillator in the harvesting device and show that, with proper design, significant performance gains can be realized as compared with a tuned linear attachment. However, we also show that the coexistence of multiple equilibria due to the nonlinearity can degrade system performance, as the system can be attracted to a low amplitude state that provides reduced harvested power. Finally, when multiple equilibria exist in the system, the basins of attraction for the stable states are determined and related to the expected response of the system.