The very interesting paper "Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1976/77), no. 3, 279–299. MR 0432626 (55 #5613), by H. Bass, E. H. Connell, and D. L. Wright, gives some answers to this question. It says that if everybody is affine and the bundle is locally trivial in the Zariski-topology, then it is a vector bundle.
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