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This paper deals with state estimation for a class of Lipschitz nonlinear systems under a time-varying disconnected communication network. A distributed observer consists of some local observers that are connected to each other through a communication network. We consider a situation where a communication network does not remain connected all the time, and the network may be caused by intermittent communication link failure. Moreover, each local observer has access to a local measurement, which may be insufficient to ensure the system's observability, but the collection of all measurements in the network ensures observability. In this condition, the purpose is to design a distributed observer where the estimated state vectors of all local observers converge to the state vector of the system asymptotically, while local observers exchange estimated state vectors through a communication network and use their local measurements. According to theoretical analysis, a nonlinear and a robust nonlinear distributed observer exist when in addition to the union of all communication topologies being strongly connected during a time interval, the component of each communication graph is also strongly connected during each subinterval. The existence conditions of the distributed observers are derived in terms of a set of linear matrix inequalities (LMIs). Finally, the effectiveness of the presented method is numerically verified using some simulation examples. [ABSTRACT FROM AUTHOR]