The isobaric multiplet mass equation (IMME) is known to break down in the first isospin quintet. In this work we combine high-resolution experimental data with state-of-the-art shell-model calculations to investigate isospin mixing as a possible cause for this violation. The experimental data are used to validate isospin-mixing matrix elements calculated with newly developed shell-model Hamiltonians. Our analysis shows that isospin mixing with nonanalog states contributes to the IMME breakdown, making the requirement of an anomalous cubic term inevitable for the multiplet.