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This paper advances a framework for evaluating mesh sensitivity in numerical models of systems involving strain-weakening materials, and demonstrates its application in the case study of the failure at the Mount Polley TSF in Canada. Numerical studies are combined with analytical arguments and physical evidence to establish that this failure was caused in part by the localisation of strain in a shear band with a thickness 0 < h sb ≤ 12.5 cm, and to rule out the possibility of slippage along a pre-sheared plane in the foundation. The range of model responses is bound by an upper limit representing the least conservative case of "no weakening," and a lower limit representing the most conservative case of strain localisation within an infinitely thin shear band, predicting the highest and the lowest stability levels, respectively. The limit states are discoverable using a new method presented here. The model's convergence is shown to vary at mesh discretisation levels above the minimum established by the upper limit state analysis; higher mesh resolutions yield lower stability levels, approaching asymptotically the floor value established by the lower limit state analysis. A conceptual relationship between a model's discretisation, the shape of the strain-weakening curve and convergence is formulated. [ABSTRACT FROM AUTHOR]