In　system-level　diagnosis,　we　propose　to　further　classify　faulty　nodes　into　two　categories.　One　category　is　the　“ordinary”　faulty　nodes　–　they　are　malfunctioning,　but　they　still　participate　in　the　diagnosis,　rendering　unreliable　test　results.　The　other　category　contains　nodes　that　are　completely　broken　down　so　that　they　cannot　test　other　nodes,　and　they　cannot　be　tested　by　other　nodes　either.　In　this　paper,　we　study　the　diagnosability　and　1-good-neighbor　conditional　diagnosability　of　hypercubes　with　both　ordinary　faulty　nodes　and　broken-down　nodes.　Let　S　be　a　set　of　missing　links　and　broken-down　nodes　in　a　hypercube　Q　n　with　|S|≤n−1.　We　prove　that　the　diagnosability　of　Q　n　−S　is　δ(Q　n　−S)　for　n≥3.　Furthermore,　we　show　that　the　1-good-neighbor　conditional　diagnosability　of　Q　n　−S　is　δ(E(Q　n　−S))+1　for　n≥4,　which　is　the　maximum　number　of　faulty　nodes　can　guarantee　to　identify,　under　the　condition　that　every　fault-free　node　has　at　least　a　fault-free　neighbor.　©　2019　Elsevier　B.V.