It is widely believed that the answer to the question in the title is in the negative.  The exclusion of such a process - tunneling of a fractionally charged quasi-particle through a potential barrier (an insulator) goes back to a "global constraint": the number of charge carriers on either side of the barrier must be (nearly) an integer.  We address the issue of a QP tunneling in connection with geometries where this constraint does not apply, point out to the feasibility of such tunneling and evaluate its amplitude.