  Balogh, Bollobas and Weinreich showed that a parameter that has since been
termed the distinguishing number can be used to identify a jump in the possible
speeds of hereditary classes of graphs at the sequence of Bell numbers.
  We prove that every hereditary class that lies above the Bell numbers and has
finite distinguishing number contains a boundary class for well-quasi-ordering.
This means that any such hereditary class which in addition is defined by
finitely many minimal forbidden induced subgraphs must contain an infinite
antichain. As all hereditary classes below the Bell numbers are
well-quasi-ordered, our results complete the answer to the question of
well-quasi-ordering for hereditary classes with finite distinguishing number.
  We also show that the decision procedure of Atminas, Collins, Foniok and
Lozin to decide the Bell number (and which now also decides well-quasi-ordering
for classes of finite distinguishing number) has run time bounded by an
explicit (quadruple exponential) function of the order of the largest minimal
forbidden induced subgraph of the class.
