Using the Penning trap mass spectrometer TITAN, we performed the first direct mass measurements of (20,21)Mg, isotopes that are the most proton-rich members of the A = 20 and A = 21 isospin multiplets. These measurements were possible through the use of a unique ion-guide laser ion source, a development that suppressed isobaric contamination by 6 orders of magnitude. Compared to the latest atomic mass evaluation, we find that the mass of (21)Mg is in good agreement but that the mass of (20)Mg deviates by 3 σ. These measurements reduce the uncertainties in the masses of (20,21)Mg by 15 and 22 times, respectively, resulting in a significant departure from the expected behavior of the isobaric multiplet mass equation in both the A = 20 and A = 21 multiplets. This presents a challenge to shell model calculations using either the isospin nonconserving universal sd USDA and USDB Hamiltonians or isospin nonconserving interactions based on chiral two- and three-nucleon forces.