CHIOS, Greece—Yasmin Ali made the perilous crossing from Turkey to this Aegean island two weeks ago even though she knew she would be trapped here, unable to travel farther into Europe.

The 19-year-old Syrian economics student is one of a rising number of people disregarding Europe’s double strategy for deterring mass migration—a deal with Turkey to return new arrivals, and the closure of Balkan borders to the north—and stretching Greece’s capacity to absorb more asylum seekers even thinner.

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