In this paper, we tackle the open problem of snap-stabilization in message-passing systems. Snap-stabilization is a nice approach to design protocols that withstand transient faults. Compared to the well-known self-stabilizing approach, snap-stabilization guarantees that the effect of faults is contained immediately after faults cease to occur. Our contribution is twofold: we show that (1) snap-stabilization is impossible for a wide class of problems if we consider networks with finite yet unbounded channel capacity; (2) snap-stabilization becomes possible in the same setting if we assume bounded-capacity channels. We propose three snap-stabilizing protocols working in fully-connected networks. Our work opens exciting new research perspectives, as it enables the snap-stabilizing paradigm to be implemented in actual networks.