Banks work by receiving deposits (which are unsecured loans that you make to the bank) and then using that money to make loans. A banks profits are the difference between the interest rate they pay on deposits and the interest rate that they charge on loans.
A bank is always going to have more money in deposits than it has cash on hand to cover all of those deposits. This is simply inherent to how banks work.
A bank run occurs when, for whatever reason, a banks depositors question the banks ability to make them (the individual depositor) whole. With a bank run, there is always a massive first mover advantage because depositors keep being able to get their money out until the bank has no more cash and all the remaining depositors are screwed.
For a very long time, this made the risk of bank runs incredibly high as everyone would pull their money on the slightest hint that the bank might not be able to cover all of its deposits immediately. And with every dollar pulled out, the odds that the bank wouldn't be able to cover increased and so the remaining depositors are at higher risk and thus more likely to pull their money.
Then we got FDIC and the federal government insuring all bank deposits up to a certain amount ($250,000 today). With that guarantee, the incentive was removed for everyone covered by that insurance to leave their money in the bank; because even if the bank collapsed, the depositor wasn't at risk any longer.
From a bank's risk management perspective, every dollar of insured deposit is basically at zero risk because the people with those insured deposits won't move their money even if the bank gets in massive trouble.
Those accounts over the insurance limit are the ones that are at risk of causing a bank run because the people owning them both have more money (thus more potential loss) and the money is at risk if a bank run starts.
SVB had something like 97% of all of its deposits uninsured. It was far and away the highest exposure of basically any bank in the entire nation. Its deposits also generally all came from the same type of customers (VC and the tech sector) and it was playing silly buggers to further increase its exposure to this relatively small sector.
Now banks have regulatory requirements to insure they have the ability to cover their deposits. A bank can stuff the money into an account with the Central Bank, but doesn't really receive any return on the money it has parked there. This is basically the banks bank account, and so banks tend to try and basically only keep as much in this account as they need to cover their expected day to day needs plus a relatively small buffer.
A bank can also meet those regulatory requirements by putting the money into T-Bill's (and a few other things that are basically T-Bill's). Since the US government is of the opinion that it will always redeem its debts on time and in full, the US government is willing to treat these holdings at full face value for those regulatory requirements (with a few caveats).
A bank can take those T-Bill's and either designate them as Hold to Maturity (HTM) or as Available for Sale (AFS). An HTM product gets treated as its full face value for various regulatory calculations while an AFS one gets treated as its current market value for those calculations. Now a bank can sell an HTM product on the market at any time, but when they do so it automatically designates a lot of the banks HTM holdings as AFS.
So say a bank needs to hold a billion dollars for regulatory reasons. It has that billion in T-Bill's that are designated HTM. Now interest rates rise and thus the market rate of those T-Bill's drops so they are only worth, say, 800 million. This doesn't matter because those T-Bill's are HTM and they will eventually become the full billion (when they mature and the US government pays them off in full).
Now the bank suddenly needs cash to cover its deposits and it has to sell, say, a hundred million of those T-Bill's. The instant it does so, that billion dollars worth of T-Bill's goes from HTM to AFS and the value of those holdings drops to 700 million. Suddenly the bank needs two hundred million more dollars to cover its regulatory requirements.
Well run banks that properly manage their assets hedge against interest rate increases and the duration of the bonds that they hold so as to avoid this issue.
Well run banks also properly manage their deposits/client base so that they have a lower risk of getting a run in the first place.
SVB failed on both ends.
The reason that they got bailed out was the risk of a broader run on the regional banks.
The really big banks, like JPM, have a de facto guarantee of all of their deposits. These are the banks that will not be allowed to fail because they are globally systematically important.
Most of the regional banks are basically one tier down and they have no such de facto guarantee. So say you are a small business who a payroll account that hovers around five million dollars at, oh, PNC. If something happens to PNC then you are screwed. But you could, instead, move that money to an account at JPM and suddenly you have basically zero risk of losing that money.
Except with every account that makes that decision the strain on those regional banks is increased and it becomes more likely that they will fail.
So SVB's depositors get bailed out to prevent all of those other uninsured accounts from running from perfectly good banks to a JPM.
At the same time, the Fed created their new instrument to let banks get cash for those HTM holdings at face value and thus not having to take the paper loss and then have to revalue their holdings.
I hate what the Fed did, but I honestly can't say that they were wrong to do it. Those SVB depositors deserved to get screwed because they were fucking idiots who fucked themselves, but if they had gotten what they deserved then there was a decent chance you would have seen the collapse of a large chunk of the regional banking sector.
Ideally, what we really need is a new insurance mechanism to guarantee larger bank deposits. Like, you pay ten thousand dollars per year and you are insured up to ten million dollars at this bank. Put a federal guarantee on it to underwrite that insure by all means, but put the insurance cost directly onto those with the large accounts.
Do that and you can mitigate a lot of the run risks.