This question came up yesterday during our index theory seminar.

Let $M$ be a 1-connected smooth manifold and let $E \to M$ be a finite-rank complex vector bundle over $M$. If all the Chern classes of $E$ vanish, what else can one say about $E$? In other words, is there an alternative characterisation of such $E$?

(For a similar question in the case of $M$ having nontrivial fundamental group, see this previous question.)