This is a mid-century Hollywood myth that just won't die. Even when it was still possible to do so, a tiny, tiny minority of Rich Americans were actually evading taxes by hiding money overseas.
Today, it's virtually impossible to hide money overseas as an American after FATCA and various other crackdowns unless you are willing to entirely give up your US citizenship. Since the $USD is the reserve currency of the world, the US government has enormous leverage in mandating foreign banks to report on the assets of every single US citizen with money abroad.
In fact, foreign banks are so scared right now, many will refuse to even deal with Americans outright. It's a huge draconian nightmare for expats--99% of whom are not rich by any means.
Ironically, if you're looking to do something nefarious, the best place to do so is actually right here in the US! Numerous states allow you to create anonymous corporations and trusts.
