Boundary of connected $n$-manifold is connected for $n \geq 2$

Question

Are there examples of manifolds of dimension at least $2$ that are connected with disconnected boundary?

Obviously the statement is false in dimension $1$ because you can just use the compact interval $[0,1]$.

Answer 1

You can turn your $1$-dimensional counterexample into a counterexample in any dimension. Just take a product of any compact manifold without boundary with $[0,1]$ (think cylinders).