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Diagnosability is an important factor in measuring the reliability of an interconnection network, while the (node) connectivity is used to measure the fault tolerance of an interconnection network. We observe that there is a close relationship between the connectivity and the diagnosability. According to our results, a t-regular and t-connected network with at least 2t + 3 nodes is t-diagnosable. Furthermore, the diagnosability of the product networks is also investigated in this work. The product networks, including hypercube, mesh, and tori, comprise very important classes of interconnection networks. Herein, different combinations of t-diagnosable and t-connected are employed to study the diagnosability of the product networks.