The opposite of a trivial truth is false; the opposite of a great truth is also true.History (or at any rate a brief google), does not record whether Bohr considered the above statement itself to be a great truth.
There is, however, another phrasing The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth. which, whilst it may initially appear to be less paradoxical than the first version, is in my opinion actually more so (or at the very least, more interestingly so). If you disagree, then tell me - what is the opposite of 'may well be'?
It cannot, surely, be 'may well not be'. because that is just a restatement of the same truth with different emphasis: 'It may well rain tomorrow. It may well not rain tomorrow.' - nobody thinks they are saying opposite things when they utter those two sentences (or, at least, not contradictory things. Are opposites necessarily contradictory? Bohr's assertion would seem to imply they are not. Contradictions are certainly not necessarily opposites (my hat is red/my hat is yellow)).
So is the opposite of 'may well be' 'is not'? It may well be(!), but if the opposite of a great truth is a great truth, it follows that the opposite of the opposite of a great truth is also a great truth and the opposite of 'is not' must be 'is'. So on that basis the second phrasing actually incorporates the first within it (whilst the first does not incorporate the second).
On the other hand, surely the opposite of an opposite should get you back to where you started (not (not x) = x), and since in the above case it does not, that suggests that 'is not' is not the opposite of 'may well be' and in fact that 'may well be' does not have an opposite ('is probably' does have an opposite in 'is probably not' because it is a statement of belief about future outcomes 'x is more likely to happen than not x' the opposite being 'not x is more likely to happen than x' but 'may well be' strikes me as simply saying 'there is a significant - but not necessarily more than 50% - probability that this is the case' and there isn't a clearly defined cut-off point for the meaning of 'significant').
Which leads to the question 'if something does not have an opposite, and the opposite of a great truth is a great truth, does that mean that something without an opposite cannot be a great truth? Indeed can it be a truth at all? Is 'having an opposite' synonymous with 'falsifiability', and is falsifiability a necessary (though obviously not sufficient) criterion for a statement to be true? (incidentally, one of my favourite paradoxes is the logical positivists' assertion that only statements which are falsifiable can be said to be meaningful is hoist by its own petard - by its own criterion that statement is meaningless)
It's all word-games of course, and I think the point is simply 'when you come across an assertion which appears to be greatly meaningful or resonant, it is often enlightening to consider its opposite as well'.