The complement of your problem L, call it L', is ``Given a graph G=(V,E) and an integer k, does G contain a simple path of length at least k+1'', which is the well-known LONGEST-PATH
problem. The problem L' is clearly in NP: Just guess the path, assuming there is one. (Equivalently, given a path, just verify it is indeed of length at least k+1.) Note that the problem is in coNP if and only if its complement is in NP, meaning L is in coNP.
Because LONGEST-PATH
is NP-complete, L is not in NP unless coNP=NP. (Because we believe coNP != NP, this implies no NP-complete problem can belong to coNP and no coNP-complete problem can belong to NP. See Arora-Barak book for details.)