I hope this is not trivial.

Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)

For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology classes.

Does there exists a vectorbundle $E$ of rank $n$ with $w_i=x_i$?

Of course $w_i$ is the $i$-th Stiefel Whitney class of $E$.