Nonlinear isolators with high-static-low-dynamic-stiffness have received considerable attention in the recent literature due to their performance benefits compared to linear vibration isolators. A quasi-zero-stiffness (QZS) isolator is a particular case of this type of isolator, which has a zero dynamic stiffness at the static equilibrium position. These types of isolators can be used to achieve very low frequency vibration isolation, but a drawback is that they have purely hardening stiffness behavior. If something occurs to destroy the symmetry of the system, for example, by an additional static load being applied to the isolator during operation, or by the incorrect mass being suspended on the isolator, then the isolator behavior will change dramatically. The question is whether this will be detrimental to the performance of the isolator and this is addressed in this paper. The analysis in this paper shows that although the asymmetry will degrade the performance of the isolator compared to the perfectly tuned case, it will still perform better than the corresponding linear isolator provided that the amplitude of excitation is not too large.