First you will need to be "introduced" to the bank by an existing customer. We're not talking about a referral link here, we're saying who introduces you is just as important as who is being introduced (i.e. long-standing customer vouching for an introduction gets more weight than a newbie trying to get a mate in).
Second, if the introduction is acceptable, then you will be invited to a meeting with one or more members of the bank's senior management to determine if you are a good fit (and no, "good fit" does not equal a commercial fit about determining how much they can sell you, its a "good fit" as in determining that you have your head screwed on and won't cause the bank problems).
You might think this is excessive, but at the most exclusive bank the management have genuine skin in the game (e.g. bank is purposely setup as unlimited liability legal structure, so if management were to ever fuck up, they really would FUCK UP because they have no corporate veil to hide behind).
It also means that dodgy overseas money and the nouveau riche can't buy their way in and start causing problems. A truly exclusive bank values quality of client over quantity of clients.