As an oncologist, Dr. Basch said he frequently sees patients making decisions based on financial considerations. If they face high out-of-pocket costs, “they’re of a mind-set to avoid visits, expensive treatments,” he said. “They have a fear.”

At Susan G. Komen, a breast cancer charity, more than half of the questions to the group’s helpline are about financial assistance, said Susan Brown, senior director of education and patient support. The organization and its affiliates provide modest grants, including one to Ms. Leonard to help pay for a test not covered by her insurance, and refer patients to other resources for aid.

“They have people all the time talking about stopping their treatment or delaying treatment,” even when an individual has had an abnormal screening and needs a work-up for a final diagnosis, Ms. Brown said.

When Ms. Leonard tried to talk with the hospital where she was getting treatment about her medical bills, she found the staff largely unsympathetic. “Because I had insurance, I was told I didn’t need a financial advocate,” she said. Instead, she relied on her contacts at Susan G. Komen and extensive research to find other sources of help.

About half of all covered workers in the United States are now enrolled in plans with a deductible of at least $1,000, and many must pay several thousand dollars in medical bills before their plans even start to cover their care. About 11 percent of covered workers have a deductible of at least $3,000, according to a survey of employer benefits by the Kaiser Family Foundation. Employers are increasingly offering these plans — and more frequently giving their workers no other option.

While high-deductible plans are meant to encourage people to think twice about whether a test or treatment is necessary and if it can be done at a lower price, “it’s also frankly to impede their use of these services,” said Dr. Peter Bach, the director of the Center for Health Policy and Outcomes at Memorial Sloan Kettering Cancer Center.