In this paper, we introduce the concept of the k-path-(bi)panconnectivity of (bipartite) graphs. It is a generalization of the (bi)panconnectivity and of the paired many-to-many k-disjoint path cover. The 2-path-bipanconnectivity with only one exception of the n-cube Qn (n⩾4) is proved. Precisely, the following result is obtained: In an n-cube with n⩾4 given any four vertices u1, v1, u2, v2 such that two of them are in one partite set and the another two are in the another partite set. Let s=t=5 if C=u1u2v1v2 is a cycle of length 4, and s=d(u1,v1)+1 and t=d(u2,v2)+1 otherwise, where d(u,v) denotes the distance between two vertices u and v. And let i and j be any two integers such that both i−s⩾0 and j−t⩾0 are even with i+j⩽2n. Then there exist two vertex-disjoint (u1,v1)-path P and (u2,v2)-path R with ∣V(P)∣=i and ∣V(R)∣=j. As consequences, many properties of hypercubes, such as bipanconnectivity, bipanpositionable bipanconnectivity [18], bipancycle-connectivity [12], two internally disjoint paths with two given lengths, and the 2-disjoint path cover with a path of a given length [21], follow from our result.