The behavioural scientist who requires an estimate of narrow heritability, h2, will conduct a twin study, and input the resulting estimated covariance matrices into a particular mode of estimation, the latter derived under supposition of the standard biometric model (SBM). It is now widely acknowledged that the standard biometric model can be expected to misrepresent, in manifold ways, the phenotypic (genetic) architecture of human traits. The impact of this misrepresentation on the accuracy of h2 estimation is unknown. Herein, we aimed to shed some light on this general issue, by undertaking three simulation studies. In each, the parameter recovery performance of five modes- Falconer's coefficient and the SEM models, ACDE, ADE, ACE, and AE- was investigated when they encountered a constructed, non-SBM, architecture, under a particular informational input. In study 1, the architecture was single-locus with dominance effects and genetic-environment covariance, and the input was { ΣMZ,T, ΣDZ,T, ΣMZ,A, ΣDZ,A}; in study 2, the architecture was identical to that of study 1, but the informational input was { ΣMZ,T, ΣDZ,T}; and in study 3, the architecture was multi-locus with dominance effects, genetic-environment covariance, and epistatic interactions. The informational input was {ΣMZ,T, ΣDZ,T, ΣMZ,A, ΣDZ,A}. The results suggest that conclusions regarding the coverage of h2 must be drawn conditional on a) the general class of generating architecture in play; b) specifics of the architecture’s parametric instantiations; c) the informational input into a mode of estimation; and d) the particular mode of estimation employed. In general, the results showed that more complicated the generating architecture, the poorer a mode’s h2 recovery performance. Random forest analyses furthermore revealed that, depending on the genetic architecture, h2, the dominance and locus additive parameter, and proportions of alleles were involved in complex interaction effects impacting on h2 parameter recovery performance of a mode of estimation. Data and materials: https://osf.io/aq9sx/