We consider a two-station cascade system in which waiting or externally arriving customers at station 1 move to the station 2 if the queue size of station 1 including an arriving customer itself and a customer being served is greater than a given threshold level с₁>=1 and if station 2 is empty. Assuming that external arrivals are subject to independent renewal processes satisfying certain regularity conditions and service times are i.i.d. at each station, we derive necessary and sufficient conditions for a Markov process describing this system to be positive recurrent in the sense of Harris. This result is extended to the cascade system with a general number k of stations in series. This extension requires certain traffic intensities of stations 2,3,...,k-1 for k>=3 to be defined. We finally note that the modeling assumptions on the renewal arrivals and i.i.d. service times are not essential if the notion of the stability is replaced by a certain sample path condition. This stability notion is identical with the standard stability if the whole system is described by the Markov process which is a Harris irreducible T-process.