There were two men who famously broke the bank at Monte Carlo in the late 19th Century, both Englishmen: one was an engineer, the other was described by an MP as the greatest swindler living. This song was inspired by the second of them.
While Joseph Jagger used his engineering nous to exploit the very slight imperfections of a roulette wheel in order to overcome the house percentage and show a modest profit, Charles Wells (1841-1926) was a confidence trickster who defrauded gullible investors in England and used their money to win considerable sums playing roulette at the Monte Carlo Casino, Monaco.
Although fraud was naturally suspected, security was paramount even in those days; Wells played a very risky system staking on even chances, and got lucky. On his second visit, he also got lucky backing the number 5. Luck does not last forever though, and although he broke the bank several times on his third visit, he ended up giving it all back. In any case, the meaning of the phrase "to break the bank" is not quite as impressive as it sounds; it means simply to win every chip on the table.
Wells' first two (successful) visits to the casino were in 1891, by the following year, his exploits had become the stuff of legend, possibly assisted by clever publicity from the casino to draw in the punters (and mugs) in the expectation that they might have the same luck. In April, Fred Gilbert wrote "The Man Who Broke The Bank At Monte Carlo", which was popularized by the music hall star Charles Coborn, and boosted the Wells legend even more. However, his aforementioned third visit, in the winter of 1892, saw him not just cleaned out but arrested at Le Havre and extradited to stand trial at the Central Criminal Court. Convicted of fraud, he was sentenced to eight years imprisonment.
Though leopards do sometimes change their spots, this one didn't; Wells served another sentence in his native England, then a third in France before dying in poverty in 1926.