WASHINGTON — The Federal Reserve has given the green light to major banks in the United States to raise dividends and buy back shares, judging them in “stress tests’’ to have a sturdy enough financial foundation to withstand a major economic downturn.

But the Fed on Wednesday rejected plans by the US divisions of two European banks, Germany’s Deutsche Bank and Spain’s Santander, and gave Morgan Stanley only conditional approval because of what were considered weaknesses in its plans dealing with risks. Morgan Stanley has until the end of the year to submit a new capital plan.

Although Santander has made progress in improving certain approaches to loss and revenue projections, the Fed said the company continues to have unresolved supervisory issues as well as deficiencies in its risk management framework. Santander has a substantial retail presence in Massachusetts through its acquisition of Sovereign Bank in 2009.

The remaining 30 banks are allowed to raise dividends or repurchase shares. They include JPMorgan Chase, Bank of America, Citibank, and Wells Fargo, the four biggest US banks.

The stress tests are an annual check-up of the biggest financial institutions in the United States. This was the sixth year of the tests. Thirty-three banks were tested to determine if they have large enough capital buffers to keep lending, even if faced with billions of dollars in losses in another financial crisis or severe economic downturn.

‘‘The participating firms have strengthened their capital positions and improved their risk-management capacities,’’ the Fed’s Daniel Tarullo said in a statement. ‘‘Continued progress in both areas will further enhance the resiliency of the nation’s largest banks.’’

The tests were mandated by Congress after the financial crisis that struck in 2008 plunged the United States into the worst economic downturn since the Great Depression of the 1930s.

The announcement on the second round of the stress tests followed last week’s initial results. The regulators determined that the 33 big banks hold more capital than at any time since the crisis and are adequately fortified to withstand a severe US and global recession and continue lending.

Under the Fed’s most extreme hypothetical scenario in this year’s tests, the US economy falls into a deep recession, causing the stock market to plunge by 50 percent. Unemployment climbs above 10 percent, while housing prices drop by 25 percent, and commercial real estate prices tumble 30 percent.

In that scenario, investors would be so panicked that yields on short-term US Treasury securities would go negative — meaning even the safest of assets would still lose money.

Beyond the hypotheticals, big banks took a huge hit from Britain’s vote to leave the European Union last week. As a sector, their shares took the largest losses by far as stocks plunged in the United States and worldwide Friday, when results of the historic vote became known.

Stock prices have since recovered, but uncertainty remains. US banks have a lot to lose in Britain’s departure from the 28-nation bloc because they do a lot of cross-border business in Europe from their offices in London. They also become less profitable when bond yields fall, since that lowers interest rates on mortgages and many other kinds of loans.

Fed officials change the stress test scenarios each year to reflect the current economic climate around the world. This year’s test did include a severe recession in Britain and in the bloc of countries that use the euro currency — but not a ‘‘Brexit’’ move by Britain to leave the European Union.

The Fed said the 33 banks would sustain $385 billion in loan losses under the most dire scenario. But even with those losses, all the banks would still together hold a high-quality capital ratio of 8.4 percent.