Summary: In a matching problem between students and schools, a mechanism is said to be robustly stable if it is stable, strategy-proof, and immune to a combined manipulation, where a student first misreports her preferences and then blocks the matching that is produced by the mechanism. We find that even when school priorities are publicly known and only students can behave strategically, there is a priority structure for which no robustly stable mechanism exists. Our main result shows that there exists a robustly stable mechanism if and only if the priority structure of schools is acyclic [H. I. Ergin, Econometrica 70, No. 6, 2489–2497 (2002; Zbl 1141.91563)], and in that case, the student-optimal stable mechanism is the unique robustly stable mechanism.