Let $(E_i)_{i\in\mathbf{N}}$ be a projective system of Banach vector bundles whose limit is a Fréchet bundle of fibre type $\mathbf{F}$.
We construct a generalized bundle of frames $\mathbf{P}(E)$ of $E$ by revising entirely the classical notion and by substituting $GL(\mathbf{F})$ with an appropriate enlarged structure group.
This is imposed by the pathology of $GL(\mathbf{F})$, which renders meaningless the ordinary frame bundle.
As a result, we prove that $E$ is associated with $\mathbf{P}(E)$ and linear connections of $E$ correspond to (principal) connections of $\mathbf{P}(E)$.
In particular, the former are necessarily projective limits of connections on the bundles $E_i$.