Nonsense like "silver platter", almost certainly embellished (unless a "Barnum" or "Wonka" or some shiite was running the bank). The fundamentals, totally believable.
It's incredibly easy (still) to do certain kinds of "social engineering". Terms like "psychological sleight-of-hand" can sometimes make it a little clearer how humans just have blind spots - ways our perception works and doesn't. And, people who are used to being VERY "in control", intelligent / experienced (compared to others in room), etc., can sometimes be the easiest to manipulate in certain ways.
But, really, it boils down, sometimes, to something as simple as "how long can you keep a person talking?" Mitnick was probably in a good position to do these sorts of things - assuming the story is from after he "turned White Hat". And, in this case, the even simpler deal with the numbers is something like "oh, shoot, I had a misprint on old cards, did I give you the right one? What's the phone number on it?" Drop something abruptly like that, at some random point in a conversation, most people wouldn't think twice... Even if their current context involves a heavy dose of thinking about voices and numbers. They might easily enough realize in the morning, but, too late, by then. Further, getting bank account numbers is not necessarily hard either. Could even be as simple as "dumpster diving", back then. Did the CEO always shred every single document, with a "secure shredder" (as much as that's possible) when home? Or maybe burn everything, always?
And, in any case, you're even mixing up aspects of the story. The phone number isn't the bank account digits, it's just all the numbers from 0 through 9 (you can even get one twice, for a 10-digit [w/ area code] number).
I propose that your sureness in dismissing this story, misapprehensions about it, etc., make you an unwittingly "good mark."