When I was a kid (long ago in the previous century) and we played poker for fun or pocket change none of us knew of such a thing as an "all-in" bet. If somebody raised you and you didn't have the funds to call you had to fold. Likewise, in movies I have seen depicting poker, normally informal, unregulated games, and games taking place in the 19th century, it seems to be a common theme of somebody not being able to call a large bet and having to resort to finding some valuable heirloom on them such as a gold watch that the other players will accept in lieu of the required call amount in order to continue playing their hand.
Obviously, the lack of an all-in bet allows a player with an early lead to apply unfair pressure by forcing a player even with the best hand to fold if they are behind on chips. But in my (admittedly limited to Google) research I haven't found any definitive answers on how long the "all-in" bet has been around, or whether there was a time frame in which players unable to call a bet due to lack of chips were required to fold. Wikipedia does seem to support the idea that table stakes aren't quite as old as poker itself:
Open stakes is the older form of stakes rules, and before "all-in" betting became commonplace, a large bankroll meant an unfair advantage; raising the bet beyond what a player could cover in cash gave the player only two options; buy a larger stake (borrowing if necessary) or fold. This is commonly seen in period-piece movies such as Westerns, where a player bets personal possessions or even wagers property against another player's much larger cash bankroll.