I've been doing some data mining from the MoGH/ChipGuide about the multiple casinos / clubs named "Bank Club" around Nevada. Here's a summary --

There have been Bank Clubs in 8 cities / towns around Nevada: Ely, Fallon, Las Vegas, Lovelock, Pittman, Reno, Searchlight and Winnemucca.

Most opened in the 1930's (after legalization in 1931) through the 40's and 50's. They were open for an average of 21 years, and with a range of 41 years (in Lovelock, including Felix's Bank Club) down to a low of just 5 years (in Searchlight) and 7 years (in Las Vegas).
By the 50's and 60's, all but one Bank Club was gone; Felix's Bank Club in Lovelock lasted until 1977 and had a Winnemucca branch too.

A number of these older bank clubs had a range of chip molds used over the years. Yet for other Bank Clubs, we don't have chips on display at all on the ChipGuide.

So why the popularity of the "Bank Club" name back in the old days, and why are none around any more?
First, a banked gambling game is one in which the "house" (the casino) fades the bets of any player ... up to the table maximum posted on each game. For instance, in blackjack, the house (dealer) has a tray of chips and pays winners / collects from losers on each hand.

In back alleys or on troop ships during WWII, it was common for players to bet against each other without a "house" or "bank" to play against. So the term / concept of "banked games" was a natural for casinos to name themselves "Bank Club" or some variable. That was in the initial years / decades when casinos were legalized in Nevada. But after a time, it was just assumed that the house (casino owners) were operating as the bank, so using the "Bank" term was no longer necessary.

It's interesting that, in California, where I am, there are still "banked" games being played where an individual player with deep pockets acts as the bank, even though there's a professional dealer that deals the cards for each game. It's just the way that California legal regulations work for casinos other than Native American casinos.

Wrapping up, here's a handful of "Bank Club" chips (all from the ChipGuide) that show the range of molds and graphics for some of the Banks Clubs around Nevada from an earlier time. Enjoy.