We study models with three coupled vector fields characterized by O(N_{1})⊕O(N_{2})⊕O(N_{3}) symmetry. Using the nonperturbative functional renormalization group, we derive β functions for the couplings and anomalous dimensions in d dimensions. Specializing to the case of three dimensions, we explore interacting fixed points that generalize the O(N) Wilson-Fisher fixed point. We find a symmetry-enhanced isotropic fixed point, a large class of fixed points with partial symmetry enhancement, as well as partially and fully decoupled fixed-point solutions. We discuss their stability properties for all values of N_{1},N_{2}, and N_{3}, emphasizing important differences to the related two-field models. For small numbers of field components, we find no stable fixed-point solutions, and we argue that this can be attributed to the presence of a large class of possible (mixed) couplings in the three-field and multifield models. Furthermore, we contrast different mechanisms for stability interchange between fixed points in the case of the two- and three-field models, which generically proceed through fixed-point collisions.