A new necessary and sufficient condition for the controllability of networked linear time-invariant systems is derived, where the network topology is general and the nodes have identical higher dimensional dynamics. The condition is easier to verify, explicitly illustrating how the network topology, node-system dynamics, external control inputs, and inner interactions altogether affect the controllability of the whole networked system. Furthermore, the controllability of the specified Cartesian product networks is revisited, revealing that the necessity of the controllability criterion established in the work presented by Chapman et al., does not hold. In view of this, a modified, necessary, and sufficient condition is established. The effectiveness of the conditions is demonstrated using several examples.