I know that the property that a vector bundle on a manifold is flat is equivalent to it being a pull-back from the fundamental groupoid. Namely, there is a map $X \to P_{\le 1} X$ from $X$ to its first Postnikov truncation, and giving a flat connection on a bundle is essentially the same as writing it as a pull-back along this map.

Are there any known "higher equations" on a connection that guarantee that $V$ is a pull-back from a higher Postnikov truncation, say, that it is a pull-back from $P_{\le 2} X$? Intuitively that would mean that the vector bundle "depends only on the first 3 homotopies, including $\pi_0$", in some sense.

Edit: As mentioned in John Klein's answer and comments, the condition of being flat is stronger than being a pullback from the 1-st Postnikov filtra, since it is actually equivlent to lifting the map $X \to BG$ through the classifying space of the discrete group $G^\delta$. Thus, one can at most ask for a $\textbf{sufficient}$ differential geometric condition on a connection to guarantee that the vector bundle factor through $P_{\le 2}X$, say.

Thus, the remaining question is: Can we find such a sufficient condition on a connection, which is weaker than flatness?