There is a longstanding correspondence between singularities and simple Lie algebras, beautifully realized in string theory by probing singularities with branes. In this talk, I will be exploring what happens when this pristine, well-established paradigm fails. The offender in this story are "T-branes", non-Abelian bound states of D6-branes that defy geometric intuition. The tool of preference will be "3d mirror symmetry", a strong/strong duality between 3d SUSY field theories. We will see that, in order to understand these situations, it is necessary to appeal to a new class of d=3 N=2 non-Lagrangian field theories whose Higgs branches mimic N=4 behavior.