Breast engorgement is swelling of the breasts. It is a common condition in moms who breastfeed. It usually happens during the first week after delivery when her milk gland tissue begins to function. The swelling can stop the milk from flowing. This can make it hard for the infant to latch. Milk supply can decrease when it is not emptied from the breast. If engorgement goes untreated, the mother is as risk for an infection in her breasts called mastitis.