An interconnection network’s diagnosability is an important metric for measuring its self-diagnostic capability. Permanent fault and intermittent fault are two different fault models that exist in an interconnection network. In this paper, we focus on the problem pertaining to the diagnosability of interconnection networks in an intermittent fault situation. First, we study a class of interconnection networks called crisp three-cycle networks, in which the cnin-number (the number of common vertices each pair of vertices share) is no more than one. Necessary and sufficient conditions are derived for the diagnosability of crisp three-cycle networks under the PMC (Preparata, Metze, and Chien) model. A simple check can show that many well-known interconnection networks are crisp three-cycle networks. Second, we prove that an interconnection network S is a ti-fault diagnosable system without repair if and only if its minimum in-degree is greater than ti under the BGM (Barsi, Grandoni, and Masetrini) model. Finally, we extend the necessary and sufficient conditions to determine whether an interconnection network S is ti-fault diagnosable without repair under the MM (Maeng and Malek) model from the permanent fault situation to the intermittent fault situation.