“We live frugally, we don’t drink, we don’t smoke, we don’t party, and we live within our means,” Mr. Franks said by phone earlier this year from Nova Scotia, where friends had given them use of an empty house. “We pay all our bills, we don’t have car payments, we pay our credit cards off every month, and that seems to count against us.”

Daniel Maranci, a lawyer in Boston who represented the Franks, said the couple met the test of earning more than enough to make a living because they had enough to hire three or four Americans as waiters and to pay for their properties.

“The marginality requirement is fairly subjective,” Mr. Maranci said. “U.K. nationals are saying there has been a shift generally in the way these cases are being adjudicated, with a more draconian view of marginality.”

Over the last two and a half years, 8,468 requests for E-2 extensions have been filed, and their approval rate does appear to have dropped, according to figures provided by William G. Wright, a spokesman for the United States Citizenship and Immigration Services. So far in the 2010 fiscal year, he said, 82 percent of the applications have been approved. In 2009, 84 percent were approved, and in 2008, 91 percent. The service does not track the reasons for denial, so the extent to which marginality was a factor is not clear.

The Franks said the vagueness of the standards made them hard to meet. “Because there are no hard and fast rules, they can get you on whatever they want,” Mrs. Franks said.

Mr. Wright, who said he was speaking generally about the process and not about the Franks’ case, said that the most recent internal reviews of the decision-making process showed that adjudicators abided by proper standards in 97 percent of cases. “The adjudicators are doing their job,” he said.

Mr. Paparelli, the California immigration lawyer, said that in recent years, E-2 visa holders were being foiled by a confluence of trends, including an increased vigilance by government officials after the Sept. 11 attacks; a perception by officials that “evil people” may be using these visas fraudulently to get into the United States; a bureaucratic disinclination to take the time to examine applications by mom-and-pop operations; and immigration officers’ perceptions that local economies already hurt by the recession and job losses could not sustain more businesses.