I guess this is not limited to principal bundles, however those are my primary interest in asking this question.
Let $(P,\pi,M,G)$ be a principal fibre bundle over $M$ with structure group $G$.
Assume that there exists an indexing set $\mathbb A$, and an open cover $\{U_\alpha\}_{\alpha\in\mathbb A}$ of $M$ by local trivialization domains, such that for any nonempty $U_{\alpha\beta}=U_\alpha\cap U_\beta$ the transition functions $$ \psi_{\alpha\beta}:U_{\alpha\beta}\rightarrow G $$ are all constants.
What can we say about the bundle then? For a while I was under the impression that this means the bundle is trivial, however I doubt that's true.