  For every prime power p^n with p = 2 or 3 and n > 1 we give an example of an
elliptic curve over Q containing a rational point which is locally divisible by
p^n but is not divisible by p^n. For these same prime powers we construct
examples showing that the analogous local-global principle for divisibility in
the Weil-Ch\^atelet group can also fail.
