A container for wet wipes includes a pop-up style dispensing means formed by a rigid port which surrounds a flexible, rubber-like material or sheet having one or more slits through which the wet wipes are dispensed. The configuration of the slits, in combination with the flexibility of the rubber-like material, enables the user to reach into the container through the slits to grasp a wet wipe in the event the pop-up feature fails, either due to fallback or a missed interfold.