Simple Temporal Networks with Uncertainty (STNU) are a well-known constraint-based model expressing sets of activities (e.g., a schedule or a plan) related by temporal constraints, each having possible durations in the form of convex intervals. Uncertainty comes from some of these durations being contingent, i.e., the agent executing the plan cannot decide the actual duration at execution time. To check that execution will satisfy all the constraints, three levels of controllability exist: the Strong and Dynamic Controllability (SC/DC) has proven both useful in practice and provable in polynomial time, while Weak Controllability (WC) is co-NP-complete and has been left aside.Moreover, controllability checking algorithms are propagation strategies, which have the usual drawback, in case of failure, to prove unable to locate the contingents that explain the source of non-controllability. This paper has three contributions: (1) it substantiates the usefulness of WC in multi-agent systems (MAS) where another agent controls a contingent, and agents agree just before execution on the durations; (2) it provides a new WC-checking algorithm whose performance in practice depends on the network structure and is faster in loosely connected ones; (3) it provides the failing cycles in the network that explain non-WC.