(Edit to add: If you meant the letter in that link what it says doesn't sound like "you will definitely get your money back, you just need to file a claim first" at all:
"Balances held by customers in accounts at the Cayman Islands Branch at failure are not deposits under the Federal Deposit Insurance Act [...] Your status as a result of the Institution's failure is that of general unsecured creditor.")
The first clue about contemporary political economy could tell us that the entire SVB meltdown would be socialized by the US state. See my comments in this thread for my predictions pre-bailout: https://news.ycombinator.com/item?id=35094447
"By law, after insured depositors are paid, uninsured depositors are paid next, followed by general creditors and then stockholders. In most cases, general creditors and stockholders realize little or no recovery. Payments of uninsured funds only, called dividends, depend on the net recovered proceeds from the liquidation of the bank's assets and the payment of bank liabilities according to federal statute. While fully insured deposits are paid promptly after the failure of the bank, the disbursements of uninsured funds may take place over several years based on the timing in the liquidation of the failed bank assets. "
The acceptance of the claims means that they will share the proceeds from the liquidation of the assets of the non-acquired remains of the bank with the other general unsecured creditors.
As far as I can see, those Cayman Islands "not-deposits" will be paid just like the outstanding invoices owed to furniture distributors, etc. They'll get at least part of it but there is no guarantee about how much - and it will take time. [Of course they could (?) still decide to treat them differently. But the current outlook is not bright.]