The proof that any vector bundle $E \rightarrow M$ has a connection $\nabla$ uses a partition of unity subordinate to some cover $\mathcal{U}$ consisting of local trivialisations of $E$. It essentially exploits the fact that the trivial bundle has a connection given by the operator $d$. I've also seen a similar style of construction for proving that every Riemannian manifold has a connection compatible with its metric.
Can this be generalised? I.e. can we construct any connection using a partition of unity?
A better question might be to ask, if we have some connection $\nabla$ on $E$, is there always some local data from which we can construct $\nabla$ using a partition of unity?