The Co-operative Bank has failed a key test of how it would perform in a financial crisis by the Bank of England, and Royal Bank of Scotland (RBS) and Lloyds Banking Group were warned that their capital position at the end of 2013 was inadequate.
The tests by the Bank of England were in order to work out whether banks had enough capital to survive, for example, a 35 percent plunge in house prices.
Other worst-case scenarios tested included interest rates rising from 0.5 percent to above 4 percent, unemployment nearly doubling to 12 percent, and a 30 percent drop in the value of sterling against a basket of other currencies.
RBS's "minimum stressed ratio" - the key threshold of whether it has enough capital to survive the theoretical stresses - before planned management actions was 4.6 percent, just above the 4.5 percent the Bank of England was looking for. Lloyds came in at 5 percent, while Co-op fell far below the threshold at -2.6 percent.
Andrew Bailey, the Bank of England's deputy governor for prudential regulation, described the results as "encouraging".
"Six year after the real depth of the financial crisis in 2008, the U.K. banking system is in better shape. It's safer and sounder," he told CNBC on Tuesday.
"Capital levels are higher and it's more able to resist the severe shock we subjected it to, and that of course allows it to support the economy."
RBS and Lloyds were told that they needed to strengthen their capital position as of the end of 2013, but the Bank of England added that their situation since then has improved. As such, only Co-op has to submit a revised capital plan.
Bailey stressed it was important to note that the banks were on "recovery paths".
"Our judgment against a severe stress – which is not what we expect to happen, but is what we're trying to protect the system against – is that they have some further work to do," he added.