Edit: to answer the OP's question, my unqualified amateur opinion is that local/smallish banks will be more useful to you in a crisis than the big guys. Then again, I prefer lots of institutions to be more local.
Since I'm fresh out of college, I don't have huge piles of $$ in the bank (or anywhere else for that matter), so I'm not as concerned about moving my eggs into different baskets. I have been slowly stowing away some cash to keep handy if the banks go on lockdown or if power goes out.
One of the reasons it is difficult to yank all of your $$ out of a bank at once, is that they technically have it invested in other things (hence you getting interest.) If everyone made a run on the banks at once, they wouldn't be able to free up that much cash instantly without liquidating themselves out of existence.
The Fed can (and will) order an emergency "banking holiday" to prevent such a run on the banks if necessary. That's where having moolah on hand becomes useful.
I don't know about keeping your life's saving in the house. One robbery while you're away would be the end of you financially. Your bank account, while not always accessible in a time of crisis, is insured by the Federal Government. I don't think our gov will be dismantled and swept away anytime soon, so I'll go with bank account.