Intentional introduction of a bi-stable nonlinear restoring element to the design of point wave energy absorbers (PWAs) has been recently investigated as a means to broaden the steady-state response bandwidth of the PWA and shift it towards lower frequencies. The underlying hypothesis, which is supported by many previous theoretical investigations, is that this will help (i) reduce the sensitivity of the PWA’s response to variations in the frequency and amplitude of the incident waves, and (ii) permit the PWA to effectively harness energy from the low-frequency high-energy content of the incident wave spectrum. In this study, we investigate this hypothesis experimentally by designing, fabricating, and assessing the response behavior of a prototype bi-stable PWA (B-PWA) excited by harmonic incident waves in an experimental wave flume. We create bifurcation maps in the wave frequency and amplitude parameters’ space and use them to define an effective frequency bandwidth outside which the B-PWA generates negligible power levels. We compare the effective bandwidth of three different B-PWAs (different bi-stable restoring elements) to the traditional linear design and demonstrate that bi-stability does not seem to improve the effective bandwidth over the linear design; at least not for the three designs considered. Moreover, the difficulty of defining the effective bandwidth of the B-PWA and its sensitive dependence on the wave amplitude adds more layers of complexity, which make the process of optimizing its performance a cumbersome task. Nonetheless, a key advantage remains in the ability of the bi-stable restoring force to push the effective bandwidth of the B-PWA towards lower frequencies, where most of the wave energy is trapped.