The best time to question is when you do not have an answer. pohnpei is right on here. Questioning makes you think, and if the question does not yet have an answer, you are forced to try other means of obtaining the answers. In other words, if there is a question with no answer (or no definitive answer, i.e. "can computers think like humans") you might want to try thinking outside the box - think in ways others have not yet tried. This is often a way to discover new ideas, new possible answers, and in the end, you will at least have developed your own thinking skills or improved upon them in some way: creative, logical, etc. At most, you may come up with some novel way of looking at a problem. Plus, philosophical problems are not just limited to abstract "Why's" although those are as worthwhile as any. Philosophical inquiries can also be things like: "when is war justified?" This question eventually becomes a political one, but each person who has an opinion on the subject usually refers to philosophical and ethical arguments to state their case. And although they sometimes do so just to drop some quotes by a revered philosopher (all for show), others have really thought about the question in social and philosophical depth. In the end, the journey is the thing; you will learn more by questioning things in greater depth.
In general, things should be questioned even if it is simply to double-check their validity. Government policy should be questioned, statistical analyses, science experiment results, and of course philosophy.
You know how they say that the fun is the journey, not the destination? I think it's the same with this. Not that it's exactly fun, but that it's worthwhile.
The more we think and question, the more we help ourselves. We help ourselves understand things better than we would if we did not question in the first place. We also help ourselves become wiser and better thinkers.
So even if we don't come up with actual answers, we have helped ourselves. We have made ourselves think about important issues and become more aware of their complexities. And we have made ourselves "smarter" just by thinking.
Ignorance stems from not asking questions and not thinking about them. If you decide asking questions to which you don't have and answer if pointless you're liable to stop questioning things and just accept them you're apt to become ignorant. With this question you might wanna ask yourself, what is the point of asking a question if you do have an answer.