Of course, not everyone would agree that we have failed to learn the lessons from these times in our past. However, it is possible to infer that we have not learned anything because it is possible to say that we are reliving the problems of the 1920s and, to some extent, those of the Depression.
Some historians argue that the 1920s was a time of excessive and unchecked capitalist greed which was based on the exploitation of the workers. The ‘20s, we can argue, were economically problematic because of the great degree of inequality between the rich and the masses. This showed that the rich were making huge profits from the work done by the masses and keeping too much of those profits for themselves. The government did not do anything to stop this trend.
We can also say that the stock market bubble of the 1920s was caused by excessive credit and by excessive greed. People wanted to get rich quick. Bankers and others were willing to loan them money to allow them to speculate on the stock market. This caused a huge bubble that eventually burst.
It is possible to say that just such factors have caused our current poor economic conditions. We once again (arguably) have a large gap between the rich and the masses. We once again have had bubbles driven by greed (notably in the housing market this time) and by excessive extension of credit. These problems have arguably caused us to have serious economic difficulties just as the problems of the 1920s helped to bring about the Depression.
George Santayana said, "Those who cannot remember the past are doomed to repeat it." This has proven true in the current economic situation, as the economic conditions that led to the Great Depression have similarities with what led to the Great Recession and it's aftermath.
The Roaring 20s were called that because it was a time of almost unprecedented economic growth. World War I had done significant damage to Europe's economic capacity, leading to increased demand for US goods, since America's economy was relatively unscathed. In the 20s that profit was showing in stocks and other securities, and rising popularity of making up for lower prices with volume of sales.
In the 1990s and post 9/11 there was again immense economic growth. New mortgage rules had made it easier for less wealthy Americans to attain home ownership, and reduced taxes had encouraged job creation that sustained the mortgages.
During the 20s, there was a little undermining process going on; incomes were not growing as fast as inflation. The same thing happened 1990-2007. Because of this, consumers were losing purchasing power, and this was contributing to overproduction, because nobody was buying. The only real solutions to this were higher compensation or lower prices. Cutting production meant lost jobs, and that only starts a vicious cycle. Credit/loans can provide temporary relief, but are merely temporary if the income to pay off the loans doesn't materialize.
In 1929, and 2007 the failure of business to raise incomes or lower prices caused profits to fall as consumer spending dropped off for lack of income. Businesses panicked and cut production and jobs. This touched off a vicious cycle as consumer incomes were lost due to unemployment, and with them consumer spending, that lead to lower business revenue and lower profits, which only spawned further cuts.
Sadly, some of the same mistakes are being made in response. We tried authorizing money for companies to stop them from cutting jobs (i.e., bailouts), and executives took bonuses while continuing job cuts. We tried a Keynesian approach of tax rebates while counting on tax payments later, but that just transferred that debt that may not be paid back later (i.e., fiscal cliff, sequestration).