We consider a directed acyclic network with unit-capacity edges and n source-terminal (si - ti) pairs that wish to communicate at unit rate via network coding. The connectivity between the si-ti pairs is quantified by means of a connectivity level vector, [k1 k2 ... kn] such that there exist ki edge-disjoint paths between si and ti. In this work we attempt to classify networks based on the connectivity level. Specifically, for case of n = 3, and a given connectivity level vector [k1 k2 k3], we aim to either present a constructive network coding scheme or an instance of a network that cannot support the desired rate. Certain generalizations for the case of higher n have also been found.