The goal of this work is to explore and develop the machinery of branched bending in finite-volume hyperbolic n-manifolds as a means of explaining the flexibility of these manifolds. Here, we define branched bending deformations as deformations supported on a piecewise totally geodesic complex of (n-1)-dimensional faces meeting along (n-2)-dimensional branching loci. We establish a lower bound on the dimension of the deformation space of such manifolds containing a branched complex, and establish some facts about the Borromean rings as a special example (recovering a result of Menasco and Reid).