My personal story is that I once wrote a check on my brokerage account to pay my monthly credit card bill; that particular month it was a few thousand dollars. This was back in the day when the brokerage account mailed me my processed checks. Well, a week or two later I get two processed checks in the mail from the brokerage company, including the one to my credit card company.
The credit card company treated my bill as paid in full by the check.
The brokerage company debited my account for the other check, but as far as they were concerned the check to my credit card was never written; they didn't debit my account.
I was of course told to wait and they would figure it all out, but they never did. The back of the check had all of those mysterious numbers and bank names showing that it went through the system somehow. Obviously there was some mistake, but since both the brokerage and credit card companies were large, multi-billion dollar entities, I think they probably wrote it off as a rounding error.
Back then I had more time and vigor, so I spent about six months getting them to debit my account. I had to call them, fax them copies of both sides of the check, then send them a firm letter. I think they thought I was weird.
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I also used to work for a bank in their "problem management department" so I got to see all of the glitches. The couple that I recall:
* One night the bank ran the job to pay all the mortgages twice. Whoops. They figured that one out and fixed it during the night so thousands of customers didn't wake up to find their checking accounts mysteriously overdrawn.
* A teller processed a mortgage payment for $375 as something like $3,750,000,000. Audit procedures were in place so that the teller couldn't fix the mistake themselves, so we had to create a special accounting job to reverse the transaction. We also had to call the customer and ask them not to ask for a $3.75 billion credit balance refund. Heh.
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Finally, I did have a bank that I fired once. The bank was an interstate bank, and I had accounts with them in the state where I was attending college, including a debit card. When I graduated and moved to Idaho, I went to the analogous branch here and asked to open accounts. They refused to give me a debit card, even though I had had one and used it responsibly for two years with their sister bank in another state, and I was now a proud college graduate with (presumably) better earning potential. Fired them and haven't done business with them since.
Oh, and my Dad fired a bank once too. He had a business with numerous employees, and somehow someone at the bank ran the payroll checks against my Dad's personal checking account, which overdrew him by probably mid-five figures. Then I think they got snooty with him that he had overdrawn his account. He tried talking to the branch manager but ended up firing them.
I'm sure all banks screw up every so often. The bank that my Dad fired is the one I've used and been happy with for about 24 years now, and they've never made a mistake with me.