We consider large fluctuations and overload of servers in a network with dynamic routing of messages. The servers form a circle. The number of input flows is equal to the number of servers; the messages of a flow are distributed between two neighboring servers; upon its arrival, a message is directed to the least loaded of these servers. Under the condition that at least two servers are overloaded, the number of overloaded servers in such a network depends on the rate of input flows. In particular, there exists a critical level of the input rate above which all servers are most probably overloaded.