FORTUNE — The Federal Reserve approved the capital plans of 16 of the nation’s 18 largest banks on Thursday as part of the final leg of their required stress tests.

Ally Financial, the former finance arm of General Motors (GM), and BB&T (BBT), a regional bank based in Winston-Salem, N.C., will be barred from increasing any distributions of capital to investors. Last week, the Fed said that Ally didn’t have enough capital to survive an economic downturn.

In addition, Goldman Sachs and JPMorgan Chase, two banks that emerged from the financial crisis among the strongest in the nation, appeared to stumble in this year’s test. The Fed said it has concerns with the capital plans of those two banks. Goldman (GS) and JPMorgan (JPM) will still be allowed to buy back stock and pay dividends, but they will have to resubmit their capital plans by the end of September to address the central bank’s concerns. The Fed declined to say what those concerns were.

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Still, the stress test showed, once again, that the nation’s largest banks are in far better shape than they were going into the financial crisis, and better than even a year ago. Nearly every bank submitted a plan to the Fed to increase their dividend or buy back shares. Both moves generally boost share prices and are cheered by investors but can deplete needed capital to cover losses from loans or bad investments. Banks used to be able to up these payouts without much oversight. But one of the changes since the financial crisis is that banks now have to get these distribution plans approved by the Fed each year.

Despite the Fed’s concerns, JPMorgan said it planned to increase its dividend to $0.38 a share, from a current $0.30, in the second quarter. It also said it planned to buy back $6 billion worth of its own stock in the next year. The firm declined to say what issues the Fed had with its capital plan, but it said it planned to address the issues and resubmit its plan. “JPMorgan Chase is fully committed to meeting all of the Fed’s requirements,” said CEO Jamie Dimon in a statement.

Bank of America (BAC) said it plans to  repurchase $5 billion worth of stock in the next year. The firm will purchase an additional $5.5 billion worth of preferred shares. Still, BofA said it has no plans to boost its dividend from its current $0.01 a share in the next year.

Wells Fargo (WFC) said it plans to up its dividend to $0.30 a share in the second quarter, up from $0.25 for the first quarter, and $0.22 a year before. The firm also said it plans to increase its share buybacks in 2013, but it didn’t say by how much. Last week, Citigroup (C) said it planned to repurchase $1.2 billion worth of its shares. Goldman could not be reached for comment.

The financial health of JPMorgan and Goldman didn’t appear to significantly differ from than that of Morgan Stanley (MS). Morgan Stanley’s capital plan was approved without any conditions. It said the Fed had approved it to buy the remaining 35% of Citi’s former brokerage unit Smith Barney that Morgan Stanley doesn’t already own.

A Fed official said that the issue with Goldman and JPMorgan may have had to do with the differences in what the two banks thought they would lose in a severe economic downturn and what the Fed had projected. Goldman, for instance, said it would lose $6.6 billion. The Fed estimated $20.5 billion.

“The financial crisis showed not only that regulators needed to increase capital requirements and conduct regular stress tests, but also that firms need strong internal processes to evaluate their own capital needs based on their individual risks and circumstances,” said Fed governor Daniel Tarullo.

The biggest surprise was the rejection of BB&T’s capital plan. BB&T passed the first round of the Fed’s stress test last week and seemed to have more than enough capital to survive a severe economic downturn. But the Fed said Thursday it has significant issues with the qualitative parts of the bank’s capital plan. It did not say what those issues were.

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Another surprise was the fact that American Express (AXP) could have had its plan rejected. According to the Fed, the capital plan that American Express initially submitted would have caused the bank to fall below the Fed’s minimum capital requirements. American Express revised its plan this week, presumably cutting how much capital it would distribute to shareholders in dividends or share buybacks. That plan wound up getting approved.

Correction: An earlier version of this story said that Ally and BB&T would not be allowed to pay any dividends. In fact, the banks are allowed to pay the dividends they already promised to shareholders. They are not allowed to make increases.