Motivated by the famous Locke's Conjecture, we study the following special kind of fault-tolerance of hypercubes. Let Qm be a certain subcube of an n-dimensional hypercube Qn, i.e., 1≤m≤n, and let F be a set of vertex-disjoint subcubes of Qm. Then (a) Qm-subcube connectivityκs(Qn;Qm) of Qn is the minimum cardinality over all F's such that Qn−F is disconnected; (b) Qm-subcube fault-tolerant HamiltonicityHs(Qn;Qm) of Qn is the maximum integer k such that Qn−F is Hamiltonian for every F with |F|≤k. In particular, if every element in F is isomorphic to Qm, then κ(Qn;Qm) (resp. H(Qn;Qm)) represents Qm-cube connectivity (resp. Qm-cube fault-tolerant Hamiltonicity) of Qn. Similarly, HLs(Qn;Qm) (resp. HL(Qn;Qm)) means Qm-subcube (resp.Qm-cube) fault-tolerant Hamilton laceability of Qn. Our main results are as follows:(1) κs(Qn;Qm)=κ(Qn;Qm)=n−m for n−m≥2, i.e., by deleting (n−m−1) m-dimensional hypercube Qm or its subcubes from the n-dimensional hypercube Qn, the resulting graph is still connected.(2) Hs(Qn;Qm)=H(Qn;Qm)=n−m−1 for n−m≥2, i.e., by deleting (n−m−1) m-dimensional hypercube Qm or its subcubes from the n-dimensional hypercube Qn, the resulting graph is still Hamiltonian.(3) n−m−2≤HLs(Qn;Qm)≤HL(Qn;Qm)≤n−m−1 for n−m≥3, i.e., by deleting (n−m−2) m-dimensional hypercube Qm or its subcubes from the n-dimensional hypercube Qn, the resulting graph is still Hamilton laceable.