The biggest U.S. banks will need more than the Federal Reserve's stamp of approval before they receive an all-clear signal from the bond market.

While relative yields on their bonds narrowed to 265 basis points, the lowest since August, that's almost 1 percentage point above last year's low of 173 in April, according to Bank of America Merrill Lynch index data. The gap between spreads on industrial and financial debt has more than doubled from a year ago and credit-default swaps on the six-biggest U.S. banks are 83 basis points higher than last April.

Fixed-income investors are showing they're not convinced the worst is over from Europe's sovereign-debt crisis and litigation from faulty mortgages issued during the housing boom, even after the Fed said 15 of the 19 biggest U.S. banks could maintain adequate capital levels in a recession scenario. Bondholders also remain concerned that increased regulations and a slow economic recovery may diminish banks' profitability.