A dispenser for selectively dispensing first and second rolls of sheet material respectively wrapped around first and second cores is provided. The dispenser includes a housing defining a storage compartment for the first and second rolls and an opening, a first and second core support structures disposed in the housing for respectively supporting differently sized first and second cores and a divider slidably carried by the housing and covering a portion of the opening. The divider is movable between a first dispensing position, wherein a first portion of the opening is uncovered to allow access to the first roll and a second portion of the opening is covered to prevent access to the second roll, and a second position, wherein the first portion of the opening is covered and the second portion the opening is uncovered to allow access to the second roll. The divider includes an adjustable blocking plate for preventing movement of the divider from the first dispensing position to the second dispensing portion until substantially all the sheet material has been exhausted from the first roll. The plate is positionable on the divider in a plurality of positions relative to the first core support structure to compensate for different diameter cores.