Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result:

Theorem: Let $f:M\rightarrow N$ be a surjective smooth map. Let $x\in N$ be a regular value of both, $f$ and $f|_{\partial M}$. Then $f^{-1}(x)$ is a manifold with boundary of dimension $m-n$ ( note that $f^{-1}(x)$ is never empty since $f$ is surjective). Furthermore $\partial (f^{-1}(x))=f^{-1}(x)\cap\partial M$.

So I'm especially interested by the last part of the theorem. So if we remove the assumption that $x$ is a regular value of $f|_{\partial M}$ then there is no reason for the theorem to hold true.

Q1: What are interesting families of manifolds (over a fixed $N$ that we always assumed to be without boundary) for which $\partial (f^{-1}(x))\neq (f^{-1}(x)\cap\partial M)$?

Q2: In general do we have some set relationship between $\partial (f^{-1}(x))$ and $(f^{-1}(x)\cap\partial M)$?