Rita Scott’s grandson convinced her in mid-November to get in on the latest investing sensation and buy bitcoin. “I thought it was a big coin,” the 70-year-old said. “I didn’t even know what it was, a piece of coin? Why would I invest in a piece of coin?”

With a few hundred dollars of her money invested in it, Ms. Scott quickly caught on and started checking the price several times a day, even while playing poker at a casino in her hometown of Las Vegas.

Late Monday,...