I recently learned the following formulation of the Darboux theorem in a class.

Theorem: Suppose $\omega_t$ is a smoothly varying family of symplectic forms on a closed manifold $M$ such that the cohomology class of $\omega_t$ is independent of $t$. Then there is a smoothly varying family of diffeomorphisms $F_t$ of $M$ such that $F_0$ is the identity and $F_t^* \omega_t = \omega_0$.

The classical formulation of the Darboux theorem is obtained as a corollary in the following way. The condition that a closed 2-form is nondegerate is an open condition, so given a symplectic form $\omega_0$ on $M$ there is a neighborhood in the space of representatives of the cohomology class of $\omega_0$ which consists only of nondegenerate forms. This neighborhood can be taken to be path connected, so the theorem guarantees that for any symplectic form $\omega_1$ which is sufficiently close to $\omega_0$ and which belongs the same cohomology class there is automatically a diffeomorphism which pulls back $\omega_1$ to $\omega_0$.

My question: is the assumption that $\omega_1$ is sufficiently close to $\omega_0$ really necessary? In the proof that I learned, we really do need a path $\omega_t$ of nondegenerate forms because the idea is to use the Poincare lemma to write $\omega_t = \omega_0 + d \beta_t$ and then obtain a time dependent vector field $X_t$ satisfying $\iota_{X_t}\omega_t = -\frac{d}{dt}\beta_t$. To say that $X_t$ exists we need $\omega_t$ to be nondegenerate for all time, and the diffeomorphisms $F_t$ are obtained from the flow of $X_t$.

I guess the real problem here is that I don't know all that many interesting examples of symplectic manifolds to begin with, and even on those examples that I know I can never produce more than one symplectic structure. Can anyone help?