It’s important to note the distinction between an investment bank and a commercial bank.
As an investment bank, the assets Lehman Brothers managed for people (stocks, bonds, etc.) are not guaranteed to hold their value but they are insured Securities Investors Protection Corp. If for some reason Lehman Brothers had not been able to transfer the assets of an account holder over to another firm the SIPC would have insured the account holder for additional losses up $500k. An investor might still suffer significant losses due to market conditions.
As it turns out the only people who lost money during the Lehman Brothers collapse were people who’d lent money to Lehman Brothers or had equity in Lehman Brothers. The actual assets managed by Lehman Brothers as a brokerage were all transferred over to Barclays Capital.
Commercial bank failures are handled in a similar way. If your bank has ever changed names unexpectedly it’s likely because it failed. When a commercial bank fails the FDIC comes in and first tries to find a buyer. As far as I know they’ve always been successful at this. All the banks assets and debts are assumed by the buyer and the banks depositors just see the letter head on their account statements change. If the FDIC we’re unable to find a buyer then each depositors would be insured $250k per account category per institution.
If you’re really worried about it, don’t keep more than $250k at a single bank.
Here are links: