Still, there’s nothing in the law that stops health care providers from billing consumers for the balance, which is what often happens — and exactly what the D’Andrea family experienced. “This is not an issue that the Affordable Care Act fixes,” said Timothy S. Jost, a professor at the Washington and Lee University School of Law and expert on health care laws. “It is conceivable that the problem gets worse for some people if the Affordable Care Act encourages narrower networks, which some people think it might do.”

If consumers use providers within their network, their annual out-of-pocket costs will vary depending on the type of plan they choose (bronze, silver or gold, for example) and their household income. But the new law states that those expenses cannot exceed $6,350 for individuals and $12,700 for a family of two or more in 2014. (This is true for all plans bought on the exchange, and for many individual plans.)

So what sort of recourse do consumers have when they receive big bills from out-of-network providers? The first step is to make sure your insurance plan paid what it should have, since many plans provide some coverage for out-of-network services. And if you were treated for an emergency service (in an emergency room or an extension of it), make sure the insurer paid at least what they would have paid to an in-network provider, said Cheryl Fish-Parcham, deputy director of health policy at Families USA, a Washington consumer advocacy group. Consumers should also check to see if their state has any additional protections against balance billing. Finally, consumers should talk to their providers about reducing any remaining charges.

Consumers would do well to follow the lead of Ms. D’Andrea’s mother, Livia Cooper. The couple relied on her to deconstruct and analyze the reams of invoices that arrived in their mailbox — including a collection notice — since they were busy caring for Sienna.

Ms. Cooper spent countless hours poring over the bills, trying to make sense of it all. “It was so overwhelming,” said Ms. Cooper, who had to close the women’s boutique she ran with Ms. D’Andrea so they could focus on Sienna, who is now 9 months old. “We received department bills and there could be 60 invoices on one printout. You would have a bill for $8,100 from one department or $6,500 from another department. It was hard to figure out what was covered, what wasn’t covered and what was balance-billed.”