It used to be the case that NSA hired the majority of all math graduates in the US, and were assumed to be years ahead in cryptography. Yet in the 90s, it became clear that they no longer were that - among other things, the cipher of the notorious Clipper chip was broken, and we can rule out that it was made weak on purpose because the whole point of Clipper was that they had a backdoor.

So, despite hiring the cream of the crop of math graduates, who could read the papers of free academia, but whose own result the free world could not access - they fell behind.

I have a theory explaining why. I think it's because science is an interactive process. NSA cryptographers could read papers, but they couldn't talk openly with the authors of those papers, because of secrecy demands - even asking question might indicate what they were working on. You can easily imagine them spending months on something they could have avoided by going to the original authors and getting told "Oh, we tried that for a long time, it doesn't work".

Whether that theory is right or not, cryptography is a concrete example of a domain where public research with fewer resources beat private research with a lot more resources.