In a competitive auction where you and partner are considering 3NT,a cue-bid of their suit asks partner to show a PARTIAL stopper if he can(of course he bids 3NT with a full stopper as well). The idea is to diagnose when the partnership together has a stopper, but neither does individually (e.g. Qx opposite Jxx etc.). Many confuse Western Q's as asking for a FULL stopper, but that is not the original intent of the convention.