It can, but this depends largely on what type of hormone replacement therapy you take and how long you take it.
There are two types of HRT: estrogen and a combination of estrogen and progestin. Because estrogen use raises the incidence of uterine cancer, the use of estrogen alone is prescribed primarily for women who no longer have a uterus. The estrogen/progestin combination therapy is usually reserved fro women who still have a uterus.
The latter combination of hormones has been shown to increase a woman's risk of breast cancer more significantly than estrogen alone. The Breast Cancer Demonstration Project found that women's risk of breast cancer went up by about 1% percent for every year that they took estrogen alone. The same study showed that, for every year that women took estrogen plus progestin, their risk went up by about 8%.
The real risk with HRT is prolonged use -- the longer a woman uses HRT, the greater her risk. For example, if a woman takes estrogen with progestin for ten years, her risk of breast cancer will be 80% higher than if she had never used hormones. For both types of therapy, however, this increase in risk begins to drop after hormone use stops and eventually returns to that of someone who has never used hormones.
Women should not take HRT for prolonged periods of time. There are many wonderful alternatives that can help women alleviate undesired symptoms of menopause, without increasing her breast cancer risk. These alternatives include natural hormone replacements, lifestyle changes, and dietary modification.