Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection.
What is a geometric interpretation of the opposit situation: The situation that the distributation is totally non integrable in the sense that the Lie algebra generated by all locall vector field tangent to the distribution is the whole lie algebra of local vector fields on $TM$
