The present study is concerned with the capability of a coupled neutron-kinetic/thermal-hydraulic code system RELAP5/PARCS for the numerical prediction of global core stability condition and instability transients. The work is motivated by the need to assess the safety significance of a number of stability transients that trigger core instability and challenge reactor protection systems. The technical approach adopted is done both to learn from real stability events and to perform analysis of idealized well-defined transients in a real plant and core configuration. In this paper, we show that the code system can serve as a unique and powerful tool to provide a consistent and reasonably reliable prediction of stability boundary even in complex plant transients. However, the prediction quality of the instability transients, i.e., core behavior without scram - namely, parameters of the limit cycle - remains questionable. We identify two main factors for future studies (two-phase flow regimes in oscillatory flow and algorithm for effective grouping of thermal-hydraulic channels) as key to enhancing the predictive capability of the existing coupled code system for boiling water reactor stability.