My technician recently told me, just before sending me gliding through an MRI tube, that MRI scans were once an uncommon breast exam. He performed the breast scans only a few times per month, when radiologists needed to get a better read on potential breast cancers than they could from mammography alone.

Now my tech performs the highly sensitive breast exam, which uses magnetic and radio waves to create layers of black and white breast images, all the time. He told me his MRI suite—with...