  In this paper we study whether symplectic toric manifolds are symplectically
cohomologically rigid. Here we say that symplectic cohomological rigidity holds
for some family of symplectic manifolds if the members of that family can be
distinguished by their integral cohomology rings and the cohomology classes of
their symplectic forms. We show how toric degenerations can be used to produce
the symplectomorphisms necessary to answer this question. As a consequence we
prove that symplectic cohomological rigidity holds for the family of symplectic
Bott manifolds with rational symplectic form whose rational cohomology ring is
isomorphic to $\mathrm{H}^*((\mathbb{CP}^1)^n;\mathbb{Q})$ for some $n$. In
particular, we classify such manifolds up to symplectomorphism. Moreover, we
prove that any symplectic toric manifold with rational symplectic form whose
integral cohomology ring is isomorphic to
$\mathrm{H}^*((\mathbb{CP}^1)^n;\mathbb{Z})$ is symplectomorphic to
$(\mathbb{CP}^1)^n$ with a product symplectic structure.
