Ok, maybe I'm not getting something. The following is true for both add-ons and rebuys, but I'll use "rebuys" for ease of typing:
If rebuys are Not allowed, then you cannot rebuy anyone. Attempting to do so will tell you that "Rebuys are not allowed", and give instructions for configuring the tournament so that they are allowed.
If rebuys are Allowed, but the current tournament conditions cause them to be disabled (such as being past the last round in which rebuys are allowed, or the maximum number of rebuys has already been reached), attempting to rebuy will give you a dialog that warns you that rebuys are not allowed and gives the reason(s) (past the last round in which rebuys are allowed, for example). This dialog has 2 buttons: Cancel and Override. If you choose to Override, you will be allowed to purchase rebuys even though the tournament conditions should prevent it.
So, it sounds like, from both descriptions from Bean_D and Phaze, that the last round in which rebuys are allowed ends and the clock pauses. However, the TD actually moves the state to the beginning of the next round and then pauses the clock. This in effect disallows rebuys, but you would like to purchase them anyway, because the next round really hasn't started yet.
Why is it that you can't press the "Override" button to accomplish this?