Investors are still worried about a run on deposits of some ailing banks while massive lifelines buy more time

• Shares of First Republic and Credit Suisse continued to sell off despite massive lifelines.
• While tens of billions of dollars bought some time, there are still doubts that depositors will stay.

Massive lifelines to First Republic and Credit Suisse spurred brief rallies before shares resumed their free falls as doubts remain that they can keep depositors over the long run.

First Republic is receiving $30 billion in deposits from Wall Street giants and large regional banks. And that's after getting $70 billion in liquidity from the Federal Reserve and JPMorgan last Sunday. Meanwhile, Switzerland's central bank provided Credit Suisse with $54 billion in liquidity.

For its part, the Federal Reserve lent $165 billion to banks in the week that ended Wednesday via its discount window and a new emergency lending program created after SVB's collapse. Despite the hundreds of billions of dollars in aid, the bank sector still sold off.

"They're funded in terms of capital. But again, once it's perceived that there's an unraveling going on — yeah, people take their money," LPL Financial's Quincy Krosby told Insider.

Jamie Cox of Harris Financial Group cautioned against relying too much on bank stocks as indicators of overall confidence, saying the lifelines were designed to protect depositors and not shareholders.

The best way to shore up faith in banks is to avoid another failure within the next week, he told Insider.

"The government has bought back some time to for banks to demonstrate that they are not going to go out of business," Cox said.

Deputy Treasury Secretary Wally Adeyemo told CNBC on Friday that deposit flows at small and regional banks have stabilized over the past week. And First Republic reported slowing deposit outflows.

But Treasury Secretary Janet Yellen acknowledged to Congress on Thursday that, unlike SVB and Signature Bank, not all deposits exceeding the FDIC's $250,000 insurance limit will be guaranteed in the future.

And that's part of the problem: because regulators declared SVB and Signature Bank systemically important and guaranteed all their deposits, other lenders — and their customers — were left wondering whether they would receive similar treatment, according to former FDIC Chairwoman Sheila Bair.

"It creates uncertainty, and the ones that do not get the bailout are under more pressure. What we're seeing with First Republic — their uninsured depositors are uncertain," she said on CNBC Friday.

If the situation worsens, then regulators should get authority from Congress to insure all deposits above the $250,000 limit across the US banking system, Bair added.

In the case of Credit Suisse, depositors have been fleeing since well before SVB failed. Last year, deposits fell by more than $150 billion, with about $100 billion of that in the fourth quarter alone.

And the lifeline from Switzerland's central bank may not be enough to reassure skittish clients.

"Whether depositors are sufficiently reassured to stem outflows over the next few days is a key question, in our view," Frédérique Carrier, head of investment strategy for RBC Wealth Management, told Reuters.

Bank mergers could help, and UBS is in talks to acquire all or parts of Credit Suisse, according to reports. First Republic is also reportedly exploring its options, including a sale.

Krosby said takeovers would not necessarily signal bad news for the bank industry.

"When you see another company coming out and saying, 'Well, we'll buy that,' you know, there's value there. It lends support," she said. "Not just for that one company that's under pressure, but for the sector."