A communicating system is k-synchronizable if all of the message sequence charts representing the executions can be divided into slices of [Formula: see text] sends followed by k receptions. It was previously shown that, for a fixed given k, one could decide whether a communicating system is k-synchronizable. This result is interesting because the reachability problem can be solved for k-synchronizable systems. However, the decision procedure assumes that the bound k is fixed. In this paper, we improve this result and show, for mailbox communication, that it is possible to decide if such a bound k exists.