Suppose $M$ is manifold with submanifolds $A$ and $B$ whose intersection is a single point $x$. Suppose further that $M$ can be written as a fiber bundle over $A$ with $B$ being the fiber at $x$, and also that $M$ can be written as a fiber bundle over $B$ with $A$ being the fiber at x.
Does it follow that $M = A \times B$?
The Klein bottle $K$ is a counterexample; since $K$ is a fiber bundle over $\mathbb{S}^1$ with fiber $\mathbb{S}^1$, but not diffeomorphic to $\mathbb{S}^1\times\mathbb{S}^1$, it will suffice be able to find submanifolds $A$ and $B$ of $K$ each diffeomorphic to $\mathbb{S}^1$ whose intersection is a single point. We can take $A$ and $B$ to be the centered horizontal and vertical lines going through the following representation of the Klein bottle: