Elon Musk's track record for technological feats as chief of SpaceX has turned skeptics into believers in everything from his quest to open space travel to Mars to his desire to build a tunnel for high-speed travel between New York and Washington. As Tesla's CEO, his ambitious vision for electric cars has also earned him a faithful following.

But now Wall Street is taking a more practical tone, increasingly questioning whether Musk can turn Tesla profitable as it burns through piles of cash and racks up debt. The company may ultimately be forced to sell new shares of its stock or take on more...