Am I stupid? I can't fit these two concepts into any type of harmony.

These types of questions may seem interesting thought exercises, but really are not. For one, the concept of "infinite time" already makes a supposition that is not reality-based (certainly not for organic poker-playing human lifeforms). So, to project any human activity to infinity is already a flawed analysis. Second, when one says "infinity" what are you really saying? It can't be logically handled because its so abstract that common concepts fail. If some event is truly infinite, then you should think that every possible outcome will eventually be realized simply because it has non-zero probability over an infinite period. Thus, if your bankroll is finite, there is some non-zero probability you bust and an infinite time to realize that non-zero probability sequence. But as you increase in time, that probability goes closer and closer to zero since your finite starting point is generally increasing. So there's an irreconcilible paradox that every sequence should eventually occur, but the sequence becomes more and more improbable as you go towards infinity. It would be an easier question if you had an infinite number of starting bankrolls, then yes, eventually a number of them would bust. The nature of "infinity" discussions will always have these paradoxes when a single trial is considered.