Everyone should maintain money accounts in two different institutions. The banks do have computer problems which take their machines offline for several hours at a time.
Furthermore, I keep investment accounts at two different brokers. One whose headquarters is in Boston and the other in San Francisco. In the case that there is a major disaster, at least half of my funds should be available.
Finally, keep some paper backup. If you get all of your account statements via email. Print summaries at least once every few months.
I have set up and tested disaster recovery systems for several European banks and government agencies. During one test we took the bank’s ATM’s in the French Antilles offline unannounced in the middle of their night so we could complete a disaster recovery test.
Come payday, I'll be opening another one at a local bank. I have to keep some relationship with BoA because they're the only bank that's local to my job, my parking place, and my home, but I'll be cutting way back on the business I do with them.