A structured linear system is an abstraction of a dynamical system which captures only the existence of interconnections between states without regard for the value of the parameters that characterize these interconnections. Such systems are structurally controllable if for almost all choices of these parameters - more precisely, for all but a set of Lebesgue measure zero in the space of parameters - it is controllable in the classical sense (using the Kalman rank condition). In this work, we bridge the gap between controllability in the classic sense and structural controllability by characterizing the space of uncontrollable parameter values for a system that is structurally controllable. Then from a network point of view, we identify relationships between network structure and the extent to which structural and classical controllability coincide.