WASHINGTON — The Federal Reserve approved plans by major U.S. banks to raise dividends and buy back shares, but gave only conditional approval for Bank of America, citing weaknesses in the bank’s planning processes.

The Fed also rejected plans by the U.S. divisions of two European banks, saying their planning for financial risks is inadequate. Those divisions belong to Germany’s Deutsche Bank and Spain’s Santander.

The Fed approvals announced Wednesday followed part of its “stress tests” — an annual check-up of the nation’s biggest financial institutions. This year, 31 banks were tested to determine if they have large enough capital buffers to keep lending through another financial crisis and severe economic downturn.

The remaining 28 banks can raise dividends or buy back shares. They included JPMorgan, Citigroup, and Wells Fargo & Co., which with Bank of America are the four biggest U.S. banks by assets.

Shortly after the Fed announced its results Wednesday, a handful of banks began increasing their dividends, including JPMorgan Chase and Wells Fargo. Bank of America announced plans to buy back $4 billion in stock after the conditional approval from the Fed.

Bank of America, the second-largest U.S. bank, has until Sept. 30 to resubmit its capital plan. If the new plan for its financial management processes isn’t acceptable to the Fed, the regulators said they may restrict dividend increases or share buybacks.

In stress tests in 2013, JPMorgan Chase & Co. and Goldman Sachs were required by the Fed to resubmit their capital plans.

This year, Wall Street powerhouses JPMorgan, Goldman and Morgan Stanley revised their capital plans to reduce their dividend payouts or share repurchases late last week after the initial results of the stress tests.

The notable “pass” among the 31 banks was Citigroup. The bank did not pass last year and Citi CEO Michael Corbat was under tremendous shareholder pressure to make sure Citi got through. Industry analysts had said previously that if Citi did not pass this time, Corbat could have lost his job.

“Citi doing as well as it did is a big positive sign for the industry as a whole,” said Nancy Bush, an industry analyst with NAB Research.

The dividends and share buybacks plans that the Fed considered are important to ordinary investors, and to banks. The banks know that their investors suffered big losses in the financial crisis, and they are eager to reward them. Some shareholders, especially retirees, rely on dividends for a portion of their income.

But raising dividends is costly, and regulators don’t want banks to deplete their capital reserves, making them vulnerable in another recession. Buybacks also are aimed at helping shareholders. By reducing the number of a company’s outstanding shares, earnings per share can increase.

It was the second straight year that the Fed rejected the plans of Santander, which is one of the biggest banks in Europe. Deutsche Bank, Germany’s biggest bank, was added to the testing roster for the first time this year.