  We make a study of Poisson structures of T*M which are graded structures when
restricted to the fiberwise polynomial algebra, and give examples. A class of
more general graded bivector fields which induce a given Poisson structure w on
the base manifold M is constructed. In particular, the horizontal lifting of a
Poisson structure from M to T*M via connections gives such bivector fields and
we discuss the conditions for these lifts to be Poisson bivector fields and
their compatibility with the canonical Poisson structure on T*M. Finally, for a
2-form on a Riemannian manifold, we study the conditions for some associated
2-forms on T*M to define Poisson structures on cotangent bundles.
