Anti-evolution people, such as creationists and intelligent design folks, often use improbability to argue that a given structure could not have evolved. These arguments are usually countered in two ways — by demonstrating a pathway whereby the structure could have developed, and by questioning the whole idea that "improbability" can even be used that way — for example, it’s often pointed out that any shuffle of a deck of cards is equally "improbable". My question deals with this second line of argument.

The counter-counter argument is that the events in question — origin of life, evolution of bird wings, etc — are not just like winning a poker game (which someone has to), but like the same person winning ten times in a row with the exact same hand, down to rank and file. At some point, say the cdesign proponentsists, you have to suspect that conventional shuffling alone is not enough, and some degree of cheating was involved.

Continuing from this, cdesign proponentsists will argue that evolution is simply unfalsifiable — if biologists can "explain" the flagellum, they can "explain" absolutely anything. Biologists counter, "What about a rabbit in the Precambrian, or a living crocoduck?" To which the more sophisticated proponentsists respond "Oh, we don’t have a problem with common descent."

So my question is simply this: Could there ever, hypothetically, be an organism or structure which was "too improbable" or otherwise highly problematic for evolution as we know it, but which did not violate common descent? And if not, could that in some sense be a problem for the testability of current evolutionary theory?

All I can think of would be an observation of saltation, like a cat giving birth to a puppy. But it could be argued that that would violate common descent. (And in this sense, perhaps the whole question is unfairly stacked against the biologists to begin with.)