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Adele_L comments on Open thread, Nov. 10 - Nov. 16, 2014 - Less Wrong Discussion
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Seems unlikely, given the existence of things like quines, and the fact that self-reference comes pretty easily. I recommend reading Godel Escher Bach, it discusses your original question in the context of this sort of self-referential mathematics, and is also very entertaining.
Quines don't say anything about human working memory limitations or the amount of time a human would require for learning to understand the whole system, and furthermore only talk about printing the source code not understanding it, so I'm not sure how they're relevant for this.
I wouldn't be too surprised if the hypothesis is true for unmodified humans, but for systems in general I expect it to be untrue. Whatever 'understanding' is, the diagonal lemma should be able to find a fixed point for it (or at the very least, an arbitrarily close approximation) - it would be very surprising if it didn't hold. Quines are just an instance of this general principle that you can actually play with and poke around and see how they work - which helps demystify the core idea and gives you a picture of how this could be possible.