Most data center services are finished by the cooperation among the connected servers. However, the malicious attackers always try to divide the network into disconnected components to start some attacks, such as the address resolution protocol (ARP) attack, the denial of service (DoS) attack, the botnet attack, and so on. The connectivity is an excellent indicator to measure the reliability and fault-tolerant ability of the network. Whereas, the traditional connectivity and current conditional connectivity cannot well reflect the fault-tolerant performance of the network when attackers are a block or have a certain structure and the components of the remaining network still have a certain structure. Based on this fact, we propose a new measure: the double-structure connectivity, which can accurately reflect the fault-tolerant ability of the network when attackers are structured and each component of the network has a certain structure after removing the attacked servers. Meanwhile, a hypercube is a high-performance interconnection network that can also be used to design some data center networks. Therefore, we study the double-structure fault-tolerance of the hypercube and determine the double-structure connectivity of distinct structures of the hypercube. Furthermore, we propose algorithms to construct structures of attackers directly to measure the fault-tolerant ability of the hypercube under this attack. Our results can be applied not only to interconnection networks but also to some data center networks.