The treewidth of a 3-manifold triangulation plays an important role in algorith-mic 3-manifold theory, and so it is useful to find bounds on the treewidth in terms of other properties of the manifold. In this paper, we prove that there exists a universal constant c such that any closed hyperbolic 3-manifold admits a triangulation of treewidth at most the product of c and the volume. The converse is not true: we show there exists a sequence of hyperbolic 3-manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.