Think about it: if commercial banks wouldn't handle finance in the way you describe, it would be virtually impossible that they go bust.
They could still be drowned by dodgy loans. But let's just say that the risk profile would be rather different.
Fractional-reserve banking is the common practice by commercial banks of accepting deposits, and making loans or investments, while holding reserves at least equal to a fraction of the bank's deposit liabilities.
Let's say we have a 5% reserve, which is realistic if you look at EU and US. Let's say there is a new bank, and you deposit 100 euro's at that bank.
I go there to lend out $95, which is allowed because the bank keeps $5. You have a nice shop, and I buy something for $95.
You deposit this $95 at the bank, which they can lend out again keeping a 5% fraction. I go there and lend out $90, buy something at your shop, and you again deposit this at the bank. We do the same with $85 and $80.
So you now have an account with 100 + 95 + 90 + 85 + 80 = $450. Nice! Now you go to that bank and request that money in cash.... oops.
Seems weird, but this is actually how it works. Look it up if you don't believe me: fractional reserve banking!!!
So even if all central bank money would be printed in bills and coins, and would be distributed to banks, that still would be only a small percentage of the numbers on everyone's bank account.
This means that those numbers on your bank account are not "money you have now", it is "money you will have in some future, when everyone pays back their loans".
So that is what they mean with a "bank run". Everyone wants their money NOW, but that is impossible, since most of the money is lended out.
So to the parent poster, even if there is no robbery, you will have a serious problem when everybody wants their money NOW. See what happened in Greece some years ago, where nobody was able to withdraw more than 50 euro's per day, since the banks had a crisis.
Thinking the money on your bank account is yours NOW is just an illusion.