Non--homeomorphic manifolds-with-boundary having homeomorphic boundaries?

Question

What is a simple example of two topological $n$-manifolds-with-boundary $M$ and $N$ that are not homeomorphic yet whose boundaries $\partial M$ and $\partial N$ are homeomorphic?

Answer 1

Poke a (large-ish) hole in a sphere and a torus ...

Answer 2

Let $K$ be any compact topological $n$-manifold with boundary, and $L$ be any compact topological $n$-manifold without boundary. Then $K$ and $K \cup L$ have the same boundary, but different numbers of components, so they're not homeomorphic.