BOUSIGNIES, France—With nearly 30% more children than five years ago, the two-classroom elementary school in this tiny northern French village is bursting at its seams: Preschoolers bunch together in a narrow room for naps and teachers can barely pass through the crowded canteen at lunch.

But the village has struggled to cover the €1 million ($1.35 million) cost of building a new school, and budget cuts have reduced state support for the plan.

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