more than one min cut in a net flow

I know the answer to the question, but I still can’t understand. I have the max flow and I need to determine whether there is more than one min-cut. I know that I need to run BFS from s in the residual network, and from t in the residual net work (after changing each (u,v) to (v,u)). What I can’t understand: why would I get different results? Every example I am trying shows me the same: Any vertex reachable from s (not include s) is a vertex that is reachable from t. Am I wrong? and if I’m wrong, what is the reason some other case can happen?