Under what conditions could the Great Depression occur again?
It is highly unlikely the another Great Depression could happen. The reason for this is that the government is much more involved in the economy today and takes steps to prevent economic downturns from getting worse.
The Great Depression was caused by a combination of factors. There was a bubble in the stock market. There was too much inequality in wealth between the people at the top of the economy and other people. These are both similar to what happened in the recent economic crisis which was started by the popping of the bubble in the housing market.
But the difference is that there are all sorts of things the government does to protect the economy today. This is why we did not have another depression. The government stepped in to bail out the financial sector. There was no run on banks because deposits are guaranteed by the government. People who lost jobs could get unemployment benefits and old people have Social Security and Medicare.
Unless these sorts of government programs disappear, another Great Depression is very unlikely.