From an information security point of view, this would provide a little bit of security as long as you used different usernames and passwords for each account. Each account requires a separate hack to compromise as long as:
1) you use different usernames/passwords
2) the accounts cannot transfer funds between each other (this makes it a pain to set up, but if a hacker can transfer funds from your other accounts to a compromised one it basically ruins the entire purpose.)
3) you don't store the account information in an easily accessible location (use a password safe at least).
Now, in terms of financial security: the FDIC insures each deposit account you hold with an approved bank. The catch being, only one account per institution is insured (or a combination of your deposit accounts up to $250,000). Also, you're only protected from the bank failing. If someone hacks your account there's no FDIC insurance that kicks in and your bank is not legally required to refund your money.
This question also reminds me of something my grandfather always said: "Use multiple banks in multiple states." He was referring to protecting your assets from divorce settlements. We all have heard that 50% of marriages end in divorce and you lose 50% of your money. The reality is a little more detailed, 5 states (including California) don't report their divorce statistics, and you lose half of what was gained during the marriage (as long as you can document what you had before the marriage and your ex-spouse can provide reliable documentation of your current finances) and potentially any money you consistently provided to your ex-spouse.
Do you see where I'm going with this? During a divorce, the court can order your bank to reveal how much you have deposited, but they have to know where you have your accounts and out of state banks don't have to comply with a different state's court orders. When you set up a bank account, you have to provide your address, but you don't need to update it when you move and it's now easier than ever to manage accounts through electronic statements. As you go through life and move around, open new accounts (local banks/credit unions are best) and have electronic statements go to a private email account. Any money you keep in a shared account will be split and any money you consistently provide to your spouse/partner (watch out for states that recognize Common Law Marriages) will be considered when determining alimony payments.