In this paper, we investigate the fault-tolerant Hamiltonian problems of crossed cubes with a faulty path. More precisely, let P denote any path in an n-dimensional crossed cube CQ_n for n ≥ 5, and let V(P) be the vertex set of P. We show that CQ_n-V(P) is Hamiltonian if |V(P)|≤n and is Hamiltonian connected if |V(P)| ≤ n-1. Compared with the previous results showing that the crossed cube is (n-2)-fault-tolerant Hamiltonian and (n-3)-fault-tolerant Hamiltonian connected for arbitrary faults, the contribution of this paper indicates that the crossed cube can tolerate more faulty vertices if these vertices happen to form some specific types of structures.