The Band Aid would cover the cuts after we cleaned them and stopped the blood, but they would not pull the cut together. If the injury were to get bumped it would bleed again.

Why could we not have a Band Aid that was elastic on all sides. One would stretch the bandage and apply it to the wound from one direction and then stretch it in the second direction and apply again. The new Band Aid would squeeze the injury closed rather than leave it open to bleed again.