  The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over
an algebraically closed field is the least positive integer m such that D[p^m]
determines D up to isomorphism (resp. up to isogeny). We show that these
invariants are lower semicontinuous in families of p-divisible groups of
constant Newton polygon. Thus they allow refinements of Newton polygon strata.
In each isogeny class of p-divisible groups, we determine the maximal value of
isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown
to be optimal in the isoclinic case. In particular, the latter disproves a
conjecture of Traverso. As an application, we answer a question of Zink on the
liftability of an endomorphism of D[p^m] to D.
