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.