The critical current of SIS tunnel structures in which the primary cur
rent-transport mechanism is a resonant tunneling through a single loca
lized state is calculated from the Gor'kov equations. The critical cur
rent is governed by a competition between tunneling and thermal mechan
isms for the decay of a state into a localized.state. The results foun
d here agree satisfactorily with experimental data. They explain the a
nomalous proximity effect and also the properties of grain-boundary Jo
sephson junctions made of high-T(c) superconductors.