The case for a breakup goes like this: As long as the deposits of Americans are sitting in the same institutions that have vast amounts of money invested in global markets, it's inevitable that the government will have to step in to bail them out should one of their bets go horribly wrong, as many did in the financial crisis, Weill told CNBC in July.
The solution? Break up the big banks, perhaps by reinstating the Glass-Steagall Act that once drew a dividing line between investment banks and consumer banks.
Lost in the debate is exactly how such a change would impact bank customers, particularly those who get investment management and insurance services from the same banks where they have their checking and savings accounts.
How to break up banks
Determining the impact on consumers is difficult in part because "breaking up the banks" can mean a lot of different things, says Bert Ely, a banking consultant and principal of Ely & Co. Inc., in Alexandria, Va.
Simply reinstating the original Glass-Steagall Act would mostly affect the types of services banks could offer corporations and high-net-worth individuals, and would have little impact on day-to-day retail banking and investing, Ely says. But the original Glass-Steagall Act wasn't very effective at separating investment banks and commercial banks, he says.
"By the time it was repealed, Glass-Steagall had been eroded a lot," Ely says. "The bright line that supposedly had existed had gotten more blurry."
Steve Turner, a partner with financial services consulting firm Novantas, agrees.
"There was a separation between what (commercial) banks could do and what investment banks could do, but there was also a gray area around the kinds of brokerage activities that could be done," Turner says.
So far, the conversation has mostly focused on Glass-Steagall, which would have virtually no impact on consumer lending, investment management and other bank services for individuals, Turner says. But it's possible that legislators could go further to try and eliminate those gray areas.
"Are we talking about restricting what banks could do in terms of offering a broad array of financial services? Would you no longer be able to go to a bank and have a brokerage account where you can invest in stocks and bonds and mutual funds?" Turner says. "If that were restricted somehow by the way that the banks were separated, that could obviously have a big impact on the wealth management business of banks -- how they're advising some of their high-net-worth clients and providing them options."