FREMONT, Calif.—Tesla Inc. blamed “production bottlenecks” for having made only a fraction of the promised 1,500 Model 3s, the $35,000 sedan designed to propel the luxury electric-car maker into the mainstream.

Unknown to analysts, investors and the hundreds of thousands of customers who signed up to buy it, as recently as early September major portions of the Model 3 were still being banged out by hand, away from the automated production line, according to people familiar with the matter.

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