This paper formulates and solves the network reconstruction problem for linear time-invariant systems. The problem is motivated from a variety of disciplines, but it has recently received considerable attention from the systems biology community in the study of chemical reaction networks. Here, we demonstrate that even when a transfer function can be identified perfectly from input-output data, not even Boolean reconstruction is possible, in general, without more information about the system. We then completely characterize this additional information that is essential for dynamical reconstruction without appeal to ad-hoc assumptions about the network, such as sparsity or minimality.