If the dimension of the base $X$ of the bundles is less than the difference $n-k$ of the fiber dimensions then the quotient bundle is trivial. To see this it is enough to consider the case $k=1$ and then induct. An embedding of a trivial rank one bundle in a trivial rank $n$ bundle amounts to a map $X\to S^{n-1}$. If the dimension of $X$ is less than $n-1$ then such a map is nullhomotopic.