  For a curve over a global field we consider for which integers d the
d-primary part of the Brauer group can obstruct the existence of rational
points. We give examples showing it is possible that there is a d-primary
obstruction for infinitely many relatively coprime d, and also an example where
the odd part of the Brauer group does not obstruct although there is a
Brauer-Manin obstruction. These examples demonstrate that a slightly stronger
form of a conjecture of Poonen is false, despite being supported by the same
heuristic.
