A dispenser is provided for dispensing stacked folded sheets. The dispenser includes a housing configured to hold the stack of sheets within an internal storage space. A cover member is pivotally attached to the housing and is movable between an open position wherein access is provided to the storage space, and a closed position wherein the cover defines a front panel of the dispenser. A dispensing opening is provided through which the folded sheets are dispensed when the cover is in its closed position. An overfill prevention device is configured within the housing. This device includes a stop member that is movably configured with a back wall of the housing and is movable between a restricting position when the cover is in its open position, and a relief position when the cover is closed. A latch mechanism is disposed in the housing in a position so as latch the stop member in its restricting position when the cover is in the open position. An actuator is disposed on the cover at a location so as to engage and release the latch mechanism when the cover is closed. The stop member is then free to move towards its relief position whereby compressive pressure on the stack of folded sheets is relieved.