This question can be taken a lot of different ways with wealth and poverty meaning different things to different people. I actually read it completely different the first time I read it but I think now I understand what you mean.
I would say it can, but not necessarily. The wealth of one might actually be the wealth of many others. However, it can be true. It's just unusual for someone to actually become wealthy off one individual becoming impoverished. Usually one becomes wealthy off many people's spending. It's hard for money to just transfer from one to the other to the point the one losing is broke.