Mammograms can miss tumors in women with dense breast tissue. So for these patients, doctors often include a second scan — ultrasound, for example, or an M.R.I. — which are more likely to turn up cancers at early stages.

But some older patients are running into an unexpected twist. Though many women see the extra scan as a routine form of prevention, Medicare won’t pay for it, and some patients are left to pick up a hefty tab.