Another good $2$-dimensional example is torus punctured once and sphere punctured three times. These become diffeomorphic when crossed with $\mathbb R$, and they have trivial tangent bundles. More generally, if you puncture a closed genus $g$ surface $n>0$ times then the product with $\mathbb R$ depends only on $2g+n$, so in dimension $2$ you can get arbitrarily large finite $k$.

But maybe you wanted a compact (without boundary) example?

EDIT In the spirit of Igor's (3), suppose we have closed smooth $n$-manifolds $M_1$ and $M_2$ with rank $k$ vector bundles $\xi_1$ and $\xi_2$, and we want to know whether the two $n+k$-manifolds $D(\xi_i)$ (closed disk bundles) are diffeomorphic. There is the necessary condition: there is a simple homotopy equivalence $f:M_1\to M_2$ such that $TM_1\oplus\xi_1$ is isomorphic (as vector bundle) to $f^*(TM_2\oplus\xi_2)$. In some cases this is sufficient. If $k>n$ then $f$ gives an embedding of $M_1$ in $D(\xi_2)$, and the normal bundle is isomorphic to $\xi_1$, so $M_1$ embeds in $D(\xi_2)$ with tubular neighborhood $D(\xi_1)$. The h-cobordism theorem shows that the outside of this is a collar. This works as long as $n+k$ is not too small. You don't need the homotopy equivalence to be simple if you just want a diffeomorphism of open disk bundles. In the case $k=n>2$ I believe this can still be made to work: Whitney trick to make $f$ an embedding and also some care about stable vs unstable normal bundle.