Let X be a smooth variety over a number field k embedded as a degree d subvariety of {P}^nk and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer varieties and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. In the case of Kummer varieties we show, more specifically, that the obstruction is already given by the 2-primary torsion. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.