CARACAS, Venezuela — The labor pains began in her village, in the dark.

Her baby was coming, and Milagros Vásquez, 20, needed help.

With a minidress stretched over her swollen and increasingly stressed body, Ms. Vásquez braved a motorbike journey across three rivers and gripped her belly through two jolting bus rides. But arriving at the first hospital was just the beginning.

Over the course of 40 hours, Ms. Vásquez, a former high school athletic star, visited a second hospital, a third, a fourth. We have no sterile tools, they told her at one. No incubator, they said at another.

She took another bus. She slept on a bench. She cried in the street, losing count of the number of doctors who had placed their hands inside her, measuring her body’s dilation, only to tell her to leave.