A FEW weeks ago, my wife and I were at our wits’ end: our 4-month-old daughter wouldn’t sleep for more than an hour at a time at night. We had consulted books and seen our pediatrician, but nothing was working. So my wife called a pediatrician who specializes in babies who struggle with sleep problems.

The next day, he drove an hour from Brooklyn to our house. He then spent an hour and a half talking to us and examining our daughter in her nursery. He prescribed some medicine for her and suggested some changes to my wife’s diet. Within two days, our baby was sleeping through the night and we were all feeling better.

The only catch was this pediatrician did not accept insurance. He had taken our credit card information before his visit and given us a form to submit to our insurance company as he left, saying insurance usually paid a portion of his fee, which was $650.

A couple of weeks later, our insurance company said it wouldn’t pay anything. Here’s how the company figured it: First, it said a fair price for our doctor’s fee was $285, about 60 percent less, because that was the going rate for our town. Then, it said the lower fee was not enough to meet our out-of-network deductible.