So, I would say, for most people in a country, if a single bank fails, you have nothing to worry.
If many banks fail, this is another story as they insurance pool of the banks may dry out.
Greece was a nightmare people don't really understand what was going on and the fact that they couldn't use an ATM, couldn't pay their bills, but at least that was a country wide issue.
When you get a similar situation but when one out of X people is left hanging in the wind because their bank has failed it would be considerably worse since there would not be a single unified emergency solution to cover them (in greece people stuck it out, there was even bartering, local stores had credit ledgers etc, GL getting credit from your local supermarket because your DB debit card won't work anymore), and when more banks start to fail because their investors and creditors lose faith and pull out (which is what happened to LB) it would make things worse and worse.
It's less than 20 days by law in my country and almost everything is done without depositor's input -- money are transferred to another bank and you simply come there with your ID to take your money in cash or open an account in another bank. I really wouldn't mind to wait a week or two as opposed to billions paid of tax money.
http://www.baltictimes.com/news/articles/30019/
So this isn't a case where a government let a bank fail, this is the traditional bailout mechanism for failed banks - the government buys them out.
This is what Latvia did with Parex, what the Netherlands did with SNS and Fortis, what portugal did with BPN and the list goes on and on and on.
For the most part no one let a bank fail the same way that the US did with LB, and no one would really want to try that with a commercial high street bank.