Compact quantum groups are non-commutative spaces with a sufficient amount of structure to resemble classical compact groups and recover much of their behavior. Their actions on classical or non-commutative spaces capture "quantum symmetries". I will discuss a curious phenomenon whereby "sufficiently regular" classical structures do not admit genuinely quantum symmetries: a compact quantum group acting in a structure-preserving fashion is automatically an ordinary compact group. This is the case, for instance, for compact quantum groups acting isometrically on the underlying geodesic metric space of a compact connected Riemannian manifold, with or without boundary. (Joint work with Debashish Goswami.)