I think from the physical point of view, the question should be reversed. Symplectic and presymplectic manifolds naturally occur in physics because such geometric structures naturally follow from a variational formulation of the equations of motion of a mechanical system or field theory (see here). This variational character is indeed connected with quantization, but there is no need to go into that here. So the interesting question should be: when are symplectic manifolds occurring in physics actually cotangent bundles?
The answer is that happens naturally only if fairly special cases. Namely, the phase space (space of initial data with symplectic structure) of a mechanical system (some field theories can be considered as infinite dimensional mechanical systems) is a naturally a cotangent bundle when the system's equations of motion are second order in time, the kinetic term is non-degenerate and non-holonomic constraints are absent. I'm being a bit vague with the terminology, so let me briefly expand on these conditions.
If $Q$ is the configuration manifold, then the natural space of initial data for a $(k+1)$-st order ODE on $Q$ is the $k$-jet bundle $J^k(Q\times \mathbb{R})_{t=0}$, where $Q\times \mathbb{R} \to \mathbb{R}$ is the trivial bundle over $\mathbb{R}$ (time $t$) with fiber $Q$. It just so happens that, given a Lagrangian defined on $J^1(Q\times\mathbb{R})$, so that the equations of motion are second order, if the Legendre transform is well defined, it establishes an isomorphism $J^1(Q\times \mathbb{R})_{t=0} \cong TQ \cong T^*Q$. In other cases, it might be possible enlarge the configuration manifold to $Q'$ and then through a sequence of transformations on the equations of motion to show that $J^k(Q\times\mathbb{R})_{t=0} \cong T^* Q'$, but I would not necessarily call that identification natural.
If the kinetic term of the Lagrangian is singular (there is no invertible map between the canonical momenta and the velocities), then the Legendre transform is not well defined and, even in the $k=1$ case, the naive identification from the preceding paragraph fails. A cotangent bundle can make an appearance here as well, but only with constraints. That is, it is possible to extend the configuration space to $Q'$ and then ultimately the space of initial data can be identified with a quotient of a submanifold of $T^*Q'$ (submanifold defined by second and first class constraints, while the quotient is defined by the flow generated by first class constraints). So, while a cotangent bundle $T^*Q'$ is used in this construction, I would not say that the end result is naturally a cotangent bundle itself.
Finally, there are special situations where the constraints are holonomic. That is, it is possible to satisfy all constraints by simply restricting the mechanical system to a submanifold $Q''\subset Q$ of the original configuration manifold. Then, for second order equations with a well defined Legendre transform, the space of initial data is once again a cotangent bundle, $T^*Q''$. However, if the constraints are more complicated, that is, non-holonomic, the identification with a cotangent bundle once again fails.
So, as you can see cotangent bundles appear as phase spaces in physics only under special conditions. It so happens that there are many examples of simple mechanical systems that satisfy these conditions, but one does not need to go far to find examples that do not.
