EDIT: Not an example because I missed the Noetherian hypothesis.  Left here for posterity.

Let R be the ring of continous functions on the 5-sphere.  Swan's theorem implies that finitely generated projective modules over this ring are the same thing as vector bundles on $S^5$.  The tangent bundle of the 5-sphere is not trivial (it is not parallelizable, which is classical).  However, the limiting homotopy group $\varinjlim \pi_5 BO(n)$ classifying stable isomorphism classes of vector bundles is trivial, and so every vector bundle on $S^5$ is stably trivial.