We present field effect measurements on discontinuous 2D thin films which are composed of a sub-monolayer of nano-grains of Au, Ni, Ag or Al. Like other electron glasses these systems exhibit slow conductance relaxation and memory effects. However, unlike other systems, the discontinuous films exhibit a dramatic slowing-down of the dynamics upon lowering the temperature below a characteristic, sample-dependent temperature, T*. For T<T* the sample exhibits a few peculiar features such as repeatable conductance fluctuations in millimeter-size samples. We suggest that the enhanced system sluggishness is related to the current-carrying network becoming very dilute in discontinuous films so that the system contains many parts which are electrically very weakly connected and the transport is dominated by very few weak links. The results provide new insight into the underlying physics of electron glasses.