I think there is more to it than just not being able to handle it. Two aspects occurred to me:
Having a saturation point: some people just keep on spending, way beyond what they can actually get comfort and enjoyment from. When poor people have this personality, they spend their windfall and go broke. When rich people have this personality, they also spend almost all the free-floating cash they have. The difference is just their money keeps growing back. However, when their money source dries up, many rich people do indeed quickly go broke as well.
It can take a lot of money to upgrade a life (permanently): for poor people suddenly receiving lots of money, it can seem like all their problems are solved. But upgrading your lifestyle, your family, your social situation can easily swallow up millions of dollars without necessarily "wasting" anything. If you're permanently and hereditarily poor, there is a more profound form of debt than just what you owe to the bank. This debt still needs to be paid, and it can easily be more than the windfall provided.