Underinsured, as the name implies, means that while you have insurance, it's not sufficient to cover all your expenses. In many cases, those that are underinsured will qualify for various forms of public assistance, such as Medicaid. Of course, being eligible for public assistance traps you to some degree. There's a great example in the NPR story about a family who the father got a raise, which then made them no longer eligible for public assistance. Because their daughter has cystic fibrosis, the employer insurance isn't significant, and they've been racking up debt ever since he got his promotion (and lost the public assistance coverage). And, what were the options the caseworker presented to them? Cut back on their hours (both he and his wife work), which might mean forfeiting his promotion, or get a divorce. Then they could apply seperately, and the mother alone would qualify for public assistance.
Pretty extreme, but not uncommon. When I was working with SCHIP enrollments, these kind of cases were pretty common. Single mothers working in the casinos, getting offers for better jobs, but having to decline them so they could keep their insurance. In other words, the system is encouraging people to stay in poverty. Boo.