Recent studies to assess very long-term seismic hazard in the United States and in Europe have highlighted the importance of the upper tail of the ground-motion distribution at the very low annual frequencies of exceedance required by these projects. In particular, the use of an unbounded lognormal distribution to represent the aleatory variability of ground motions leads to very high and potentially unphysical estimates of the expected level of shaking. Current practice in seismic hazard analysis consists of truncating the ground-motion distribution at a fixed number (ε max ) of standard deviations (σ). However, there is a general lack of consensus regarding the truncation level to adopt. This paper investigates whether a physical basis for choosing ε max can be found, by examining records with large positive residuals from the dataset used to derive one of the ground-motion models of the Next Generation Attenuation (NGA) project. In particular, interpretations of the selected records in terms of causative physical mechanisms are reviewed. This leads to the conclusion that even in well-documented cases, it is not possible to establish a robust correlation between specific physical mechanisms and large values of the residuals, and thus obtain direct physical constraints on ε max . Alternative approaches based on absolute levels of ground motion and numerical simulations are discussed. However, the choice of ε max is likely to remain a matter of judgment for the foreseeable future, in view of the large epistemic uncertainties associated with these alternatives. Additional issues arise from the coupling between ε max and σ, which causes the truncation level in terms of absolute ground motion to be dependent on the predictive equation used. Furthermore, the absolute truncation level implied by ε max will also be affected if σ is reduced significantly. These factors contribute to rendering a truncation scheme based on a single ε max value impractical.