The rapid development of cloud computing, artificial intelligence and Internet of Things has not only led to high-speed growth of data centers, but also put forward a higher level of demand for data centers. Data center networks (DCNs), as crucial components of data centers, play significant roles in the overall operation and function realization of data centers. As a server-centric DCN, BCube connected crossbars (BCCC) offer excellent network performance in terms of great scalability, low communication latency and high robustness to component failures. As the scale of BCCC increases, so does the likelihood of server failures, it is critical to identify and replace faulty servers promptly for network reliability. In this work, we first investigate the intermittent fault diagnosability and local diagnosability of BCCC(n, k). On this basis, we also determine the traditional diagnosability of BCCC(n, k) and the fact that BCCC(n, k) has strong local diagnosability under the PMC model and MM \(^*\) model. Subsequently, we further obtain that the faulty BCCC(n, k) with \(\min \{n-1, k-1\}\) missing edges can still maintain strong local diagnosability under the PMC model. In addition, we present a corresponding local diagnosis algorithm to identify the vertex state and evaluate its performance by simulation experiments, which show that the algorithm keeps good diagnosis correctness even though the number of faulty vertices in BCCC(n, k) reaches 30%.