It is a sign of the times: A new game show, “Paid Off,” debuted Tuesday offering winners not dream vacations or new cars, but a pile of cash to help lessen the crushing weight of their student debt.

The reach of America’s student loan problem — total debt is now about $1.4 trillion — is vast. Millions of people are in default, and many young people are graduating into adulthood facing payments that limit their ability to buy homes and to start families of their own.

A new analysis of federal loan data indicates that the average student’s debt load is plateauing, and perhaps even declining slightly, at least when adjusted for inflation.

That should be welcome news, except that it comes with a major asterisk: College has not become more affordable, but more students seeking bachelor’s degrees, especially at higher-cost colleges, have borrowed as much as they can under the federal loan program.

As a result, the analysis suggests, many parents are going deeper into debt to pay for their children’s education.

For students receiving bachelor’s degrees, the average debt load at graduation was $30,301 in 2015-16, about the same as estimates for the previous three years, according to the analysis, which is based on data from the National Postsecondary Student Aid Study. The federal Department of Education’s National Center for Education Statistics conducts the study every four years.

Mark Kantrowitz, the publisher and vice president of research at SavingForCollege.com, ran the numbers. He believes the explanation for the flattening is that many students simply cannot borrow any more money through the federal loan program.

More than 40 percent of students in the 2015-16 school year reached the limit of $31,000 for dependent students, up from 39 percent in 2011-12.

But some students — those whose parents cannot get federal parental loans, called PLUS loans — have a higher cap on how much they can borrow. (They can borrow up to $57,500, the same limit used for independent students.)

The percentage of all students taking advantage of the higher maximum loan amounts rose to 7.4 percent of those receiving bachelor’s degrees in 2015-16, up from 5.8 percent in 2011-12 and 3.3 percent in 2007-08, according to Kantrowitz’s analysis.

Meanwhile, between 2012 and 2016, parents’ average debt load at graduation for federal PLUS loans rose 14 percent, or $4,090, to $33,291.

“Parents are a pressure-relief valve for when students hit the Stafford loan limits,” Kantrowitz said, referring to the federal loan program.

Among PLUS loan borrowers in 2015-16, roughly two-thirds were taking loans on behalf of students pursuing bachelor’s degrees who had reached their loan limit in their senior year, Kantrowitz said.

In a further wrinkle, although parental debt loads have risen significantly, the number of parents taking out federal PLUS loans declined by about 10 percent from 2011-12 to 2015-16.

“It may be that we are in an economic recovery and as the stock market improves, fewer parents are borrowing and don’t need to borrow,” Kantrowitz said. “But those who borrow need to borrow more because the cost of college continues to rise,” he said.