Algorithm to find whether there is a path (any path) above length X between two vertices
If you allow duplicate vertices on your path, you can make a network from your graph by giving each edge weight $-1$. Then, use Bellman Ford to find the shortest path and see if it has length $-X$ or less. 

If you don't allow repeats, then the problem is NP hard, since finding the longest simple path between two vertices tells you whether there is a Hamiltonian path or not.