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Well, Babbage never wanted to build the Analytical Engine, he said so explicitly. He just wanted to design it, and let someone else build it. The reason was his experience with the Difference Engine. Because the D.E. would be useful for accurate, error-free charts for the British Navy, he felt that the Government should pay for it. And he did get an agreement with the Gov, but getting the money out of them was like getting water out of a stone. He was so disgusted by the whole process, that he resolved neither to finish the D.E. nor to even start the A.E. Other than that, I think there is no question that the Analytical Engine could have been built in the 19th century.

He would've fit in well around here ;)

Yeah, we can. That is why I am so excited about the Logical Engine that Rob B is building at http://halfbakedmaker.org/ -- he makes concrete progress on his rod-based mechaputer practically every weekend.

Check it out, it's a well-thought-out, well-documented, interesting take on mechanical computing.

Which brings up an interesting point, would engineering methods other than the waterfall method have even worked for this? It certainly sounds like Babbage struggled with this.