Consider a real and finite-dimensional smooth vector bundle $E$ over a smooth manifold $\mathcal{M}$. Furthermore, lets consider a $C^{\infty}(\mathcal{M})$-submodule $\mathcal{D}$ of the space of sections $\Gamma^{\infty}(E)$. Does there exists a subbundle $F$ of $E$ such that
$$\Gamma^{\infty}(F)=\mathcal{D}? $$
Intuitively, I would say yes, by Serre-Swan, however, in praxis it is hard to find examples. For example, what would be the subbundle of $E$ whose sections are exactly the compactly supported sections of $E$?
There are many other counter examples apart from compactly supported sections. For example you can take sections vanishing in a point $x_0\in\mathcal M$ or on some specified subset $A\subset\mathcal M$. Serre-Swan only deals with finitely generated projective modules, which is a pretty strong algebraic restriction.