The twins asked Mr. Shrem to help them amass the beginnings of what would become an enormous stockpile of cryptocurrencies, giving him $750,000 to buy Bitcoin from other deep-pocketed investors.

A few months into this partnership, the twins said, they realized that Mr. Shrem had not given them all the Bitcoin they were due. The brothers gave Mr. Shrem $250,000 in September 2012, but the lawsuit says that a month later, he delivered only around $189,000 worth of Bitcoin at the going price, which was around $12.50.

The 5,000 or so missing Bitcoins became a point of tension between the twins and Mr. Shrem. They asked him numerous times for an accounting of the Bitcoins he had purchased and eventually brought in an accountant who documented the missing funds, according to court documents.

“I have been patient, and at this point it’s getting a bit absurd,” Cameron Winklevoss wrote to Mr. Shrem in 2013 in an email quoted in the lawsuit. “I don’t take this lightly.”

The missing Bitcoins, which were worth 98 percent less at the time, appeared to have been forgotten in a broader battle between the brothers and Mr. Shrem over an investment in Bitinstant.

In 2013, Bitinstant fell apart, and the twins blocked Mr. Shrem’s efforts to revive the company with new investors because of their concerns about his management style. By the time Mr. Shrem was arrested in 2014, as a result of activities at Bitinstant that took place before the brothers invested, they had cut off contact with him.

The Winklevoss twins’ problems with Mr. Shrem have not held them back. They were briefly each cryptocurrency billionaires last year, and they have built one of the leading cryptocurrency exchanges, Gemini. Despite this year’s big drop in cryptocurrency prices, their holdings are still worth nearly a billion dollars.