In effect the idea that we aren't talking about reality is the picture Plato had, often summed up in the image of the cave - that we are in a cave and can only study the shadows of reality on the wall of the cave, not the 'true' world that is not part of our world. Plato took this viewpoint from an arbitrary philosophical basis that the 'real' world was perfect - so, for instance the real world might contain the perfect archetype of 'dog' where in our cave we just experience a shadow of dogness.
Something closer to modern science comes out in Kant's 'Ding an sich', explored so enjoyably in Adam Roberts' science fiction novel The Thing Itself. Here, there is a (non-perfect) reality in our universe but we can never experience it. As we can only interact with it through our senses, we can will never know what it is. And that brings us on neatly to the strange case of the superposed Bohr.
In his excellent collection of essays Why Quark rhymes with Pork, physics professor David Mermin discusses the Niels Bohr quote:
There is no quantum world. There is only an abstract quantum physical description, It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.Now Mermin gets his knickers in a twist claiming this isn't a quote, as Bohr didn't write these words, but rather it was reported that he said it by Aage Petersen. Mermin uses a very odd definition of a quote that only allows the written form - the OED defines a quote as a 'quoted passage or remark' - some of the best quotes in history would have to be erased if we only accepted the (initially) written form. But it is true that reported quotes are more likely to contain errors, and the topic becomes interesting when Mermin speaks with two physicists who knew Bohr personally.
Apparently, Victor Weisskopf claimed that Bohr could not possibly have said anything like this, while Rudolf Peierls said that this was exactly the kind of thing that Bohr liked to say. So in good quantum style, Bohr appears to be in a superposition of the 'said it' and 'didn't say it' states. (For what it's worth, I think Peierls was right - Weisskopf seemed to be denying the possibility because he thought it was a ridiculous idea, rather than because Bohr wouldn't have said it.)
There is something very Ding an sich like about this statement (whoever said it). And apparently Bohr did write 'In our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience.' Which comes pretty close.
Although the original quote (or non-quote) was specifically about quantum physics, the second was much wider and reminds us of something it's easy for both working physicists and those who report science to forget. When we talk about the constituents of the atom, when we say that light is like a wave or a particle or a disturbance in a quantum field, when we speak about black holes or the big bang - these are not reality. There are real phenomena which we can indirectly observe, but we will always be dealing with models, with descriptions based on our indirect measurements and theories, not 'the real essence' as Bohr put it.
This doesn't mean we can't make huge achievements using these models. All our modern electronics depends on the effectiveness of the modelling of quantum mechanics. So, in practical terms it really doesn't matter that we aren't dealing with reality. But when considering pure science, we should never fall for the glamorous elegance of our models or the bewitching glitter of a big machine like the Large Hadron Collider. We are not exploring reality. All we can ever do is construct a better model, a better way to talk about nature. Don't get me wrong, though - it's a wonderful achievement, but often misunderstood.