We study mean dimension of shifts of finite type defined on compact metric spaces and give its lower bound when the shift possesses a certain "periodic block" of arbitrarily large length. The result is applied to shift maps on generalized inverse limits with upper semi-continuous closed set-valued functions. In particular we obtain a refinement of some results due to Banic [1] and Erceg and Kennedy [5] on the dimension of the inverse limit spaces and topological entropy of their shifts.