Elon Musk recently told attendees at the South by Southwest festival that two things are keeping him up at night: an apocalyptic future created by artificial intelligence, and production delays of Tesla Inc.’s Model 3 car.

Mr. Musk has good reason to worry about Tesla. The auto maker is entering one of the most critical phases in its history, a make-or-break period in which Tesla must boost production of the Model 3 or possibly face severe financial consequences.

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