U.S. banks are going to new lengths to ward off a surprising threat to their financial health: big cash deposits.

State Street Corp., the Boston bank that manages assets for institutional investors, for the first time has begun charging some customers for large dollar deposits, people familiar with the matter said. J.P. Morgan Chase & Co., the nation’s largest bank by assets, has cut unwanted deposits by more than $150 billion this year, in part by charging fees.

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