We show that a family of disordered systems with nonrelaxational dynam
ics may exhibit ''glassy'' behavior at nonzero temperature, although s
uch a behavior appears to be ruled out by a face-value application of
mean-field theory. Nevertheless, the roots of this behavior can be und
erstood within mean-field theory itself, properly interpreted. Finite
systems belonging to this family have a dynamical regime with a self-s
imilar pattern of alternating periods of fast motion and trapping.