By Nori's theorem, the existence of non-trivial finite vector bundles on a reduced connected scheme $X$ of finite type over a perfect field $k$ with a rational point is equivalent to the fact that for every finite group scheme $G$ over $k$, every $G$-torsor is a pullback from $k$.

On $X = \mathbb A^n\smallsetminus \{0\}$ over an algebraically closed field of positive characteristic there are non-trivial finite bundles: in this case $X$ has lots of non-trivial connected étale covers (for example, by Artin-Schreier).

In characteristic 0, however, this does not happen; it follows from the equivalence between étale covers of $X$ and of $\mathop{\rm Spec}k$, and from the fact that every finite group scheme over $k$ is étale, that there is an equivalence of $G$-torsors on $X$ and on $\mathop{\rm Spec}k$.

By Nori's theorem, the existence of non-trivial finite vector bundles on a reduced connected scheme $X$ of finite type over a perfect field $k$ with a rational point is equivalent to the fact that for every finite group scheme $G$ over $k$, every $G$-torsor is a pullback from $k$.

On $X = \mathbb A^n\smallsetminus \{0\}$ over an algebraically closed field of positive characteristic there are non-trivial finite bundles: in this case $X$ has lots of non-trivial connected étale covers (for example, by Artin-Schreier).

In characteristic 0, however, this does not happen; it follows from the equivalence between étale covers of $X$ and of $\mathop{\rm Spec}k$, and from the fact that every finite group scheme over $k$ is étale, that there is an equivalence of $G$-torsors on $X$ and on $\mathop{\rm Spec}k$.

[Edit]: Of course Torsten is right, Nori's theorem applies to projective varieties. The answer above is just wrong.