What about if the alternative is no bailout, the bank goes under, and you get almost nothing back at all?
Like it or not, when you deposit 100 euros in a bank, you don't have 100 euros anymore - you instead now have an unsecured bank debt with a face value of 100 euros.
I'm not sure what kind of scenario you're imagining here. If 10% of all deposits and 0% of all bonds (which generally take haircuts before deposits) are sufficient to keep the bank afloat, the alternative is not 100% of all deposits and 100% of all bonds. The alternative is a larger portion of bonds and a smaller portion of deposits.
All banks (at least in Poland) are backed up by kind of guarantees fund, which allows to return some money to clients in case of bankruptcy. (WRONG INFO: It's 100% for amounts < 1000 EUR, 90% for amounts between 22'500 EUR and 0% (no guarantees) above that.) EDIT: actually it's 100% up to 100k EUR and 0% above.
But it's kind of risk you should be aware of when keeping money in bank. Simple taking money from accounts under risk of bankruptcy isn't.
And what good did that 100% up to 100k EUR do people with bank accounts in Cyprus? That's the whole point. They broke the deposit guarantee. Now you really do have an unsecured bank debt.
If you have 100k EUR in a bank account in Cyprus today, you're going to get 93.250k EUR on Tuesday. If instead the bank went bankrupt you'd have 100k EUR in your bank account on Tuesday too thanks to the deposit insurance. You'd be better off if the bank actually went bankrupt!
The typical pro-bailout argument at this point is that if Cyprus lets the banks go bankrupt the deposit insurance scheme (in other words, the government) won't be able to make everyone whole. That argument is a tacit admission that bank accounts really are unsecured.
(I am from Poland, EU)