While medical reform was supposed to make health care more accessible, industry analysts say the problem persists. In 2015, for example, national health officials recommended that for critically ill patients, hospitals had to “first save them and then demand payment later.”

New parents are vulnerable, according to Chinese state media reports.

In 2012, a couple in Shenzhen were denied access to their newborn twins for two months because they could not pay nearly $19,000 in fees. In 2011, a 57-year-old grandmother in Nanjing, whose son owed a hospital $2,800 in medical fees for his newborn, knelt and begged doctors and nurses to allow him to see his child. That same year, a hospital in Dongguan told the parents who owed it more than $1,600 that it had sent their newborn child to an orphanage in order to “frighten” them into paying. While the families inevitably get their babies back, hospital officials can use their demands for faster or fuller payment.

Rebecca Taylor, an Australian breast-feeding counselor in Beijing, called Ms. Logbo’s case “a ginormous violation of human rights.” She added that separating Ms. Logbo from her babies could be “almost catastrophic” in terms of breast-milk production.

“I’m saddened, disappointed and horrified, but I’m not surprised,” Ms. Taylor said. “If anybody goes to a local hospital for anything, everybody knows you have to go to the A.T.M. first to carry a fistful of cash. You will literally not get things without paying.”

Ms. Logbo acknowledged that her situation complicated matters. Her boyfriend, also a Liberian and the father of her twins, has been detained in China since September, she said, accused of lending his Chinese bank account to a friend for a money transfer.

At the Huadu District People’s Hospital in Guangzhou, the demands for money came early. On May 5, as Ms. Logbo was going into labor, she had to pay $130 for an “ambulance fee.” After her C-section the next day, she had to pay a $790 deposit.