The primary goal of this 5-year study is to determine whether exemestane alone or in combination with celecoxib decreases breast tissue density in healthy postmenopausal women at high risk for breast cancer. Dense breast tissue seen on mammography has been linked to an increased risk of breast cancer. The study will also examine the effects of exemestane and celecoxib on bone density, blood hormone levels and quality of life. Exemestane, approved by the Food and Drug Administration for treating postmenopausal women with breast cancer, lowers the amount of estrogen in the body. Celecoxib, approved for treating arthritis pain and for reducing the number or colon polyps in an inherited syndrome, is an anti-inflammatory drug. Half of the women in the study will receive exemestane alone and half will receive exemestane and celecoxib together.