In theory, a bank can borrow reserves from/via the Fed to fulfill redemption requests, so long as the Fed and other lenders in the financial system think the bank has a sound balance sheet (i.e., its assets, including the loans it has made, are truly worth more than its liabilities, including the deposit balances it owes to depositors). The Fed can never run out of money to lend; it creates it out of thin air to lend them as needed with the push of a button.

If a bank does get in trouble, it's usually because its balance sheet is not sound (i.e., assets are not really worth more than liabilities). In the case of SVB, a big portion of its assets are loans made to startups. Who knows how many of those loans are at risk of default in this environment? Simultaneously, many of SVB's depositors are also startups that at the moment can't raise more capital and thus have been withdrawing money from their bank accounts to fund their cash burn. I suspect SVB can't sell its doubtful loans, or use them as collateral, so it has been forced to sell high-quality long-duration assets purchased when rates were much lower, recognizing large losses.