This very well might be the most amusing and rock solid logically based argument against God I've ever seen. Visit Why Won't God Heal Amputees for an exhaustive delving into this subject complete with common rationalizations and their rebuttals. It boils down to a simple hypothesis test and the inability to disprove the null despite only needing one confirmed case to do so.

Why doesn't God ever heal amputees? We all know no one has ever spontaneously regrown a limb through prayer. We also know that religionists claim that cancer is cured through prayer every day and that "if you have faith as a grain of mustard seed...nothing will be impossible to you". A single confirmed case of a regenerated limb through prayer would prove the existence of God, but history leaves us lacking.

Here is a fairly concise exerpt:

Explaining the case of amputees

Just for a moment, I would ask you to consider the possibility of another explanation. If you believe in God, then this explanation will initially appear to be complete nonsense. However, it is interesting in light of the conversation we will be having in this book.

One explanation for the evidence that we see before us is this:

God exists, and God answers prayers, but for some reason God chooses to ignore the prayers of amputees. We don't have a good explanation for why God acts this way, and it does seem to contradict what Jesus teaches about prayer in the Bible, but clearly God has his divine reasons.
Now let's look at the situation with amputees from another point of view. This explanation is more straightforward:

God is imaginary.

Let's look at what happens when we consider this explanation and see how it stacks up.

Assume that God is imaginary. The beauty of this explanation is that it fits the facts perfectly. In the case of amputees, it is a valid way to explain the reality that we see in our world. The logic goes like this:

If God is imaginary, then he does not answer any prayers. Therefore, the prayers of amputees would go unanswered too.

The thing that is so appealing about this explanation is that there is no hand waving. There are no contradictions. It is completely fair. There is no paradox. This explanation makes complete sense in light of the evidence we see in our world.

Interestingly, this explanation also happens to cover the case of Neva Rogers in Chapter 1. And Steve Homel's subdivision in Chapter 2. And Ranika in Chapter 4. If you assume that God is imaginary, then the paradox of God evaporates in all of these cases. Why did Ranika die? Because there was no all-powerful, prayer-answering God to save her. Why did Neva die? Because there was no all-powerful, prayer-answering God to save her. Why did Steve's house remain standing while 39 others burned to the ground? Because there was no all-powerful, prayer-answering God to save any of the houses (and Steve's house was a fluke). Why did 200,000 people die in the tsunami? Because there was no all-powerful, prayer-answering God to save them. And so on. It explains amputees too. The paradox of God vanishes completely.