According to an article by Amy Norton for Medline Plus, "breast cancer screening that includes MRI scans might help find cancers at an earlier stage in high-risk women, reducing the likelihood that the tumors will become advanced before they're diagnosed."
Women who have mutations in BRCA1 or BRCA2 genes are at higher risk for developing breast or ovarian cancer. Their chances for developing the disease are 60% (vs. 12% by the general population).
Therefore, the American Cancer Society advises that these high risk women be screened with both a mammography and an MRI.
The MRI screening, however, has its drawbacks. It's much more expensive than mammograms (about $1,000) and the test needs to be repeated every year. There comes a higher risk of false-positives with an MRI.
Some of this group opts for preventive mastectomy, a less expensive way to deal with a possible cancer in the future.