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I don't really know what "profound" means here, but I usually take Bohr's maxim as a way of pointing out that when I encounter two statements, both of which seem true (e.g., they seem to support verified predictions about observations), which seem like opposites of one another, I have discovered a fault line in my thinking... either a case where I'm switching back and forth between two different and incompatible techniques for mapping English-language statements to predictions about observations, or a case for which my understanding of what it means for statements to be opposites is inadequate, or something else along those lines.
Mapping epistemological fault lines may not be profound, but I find it a useful thing to attend to. At the very least, I find it useful to be very careful about reasoning casually in proximity to them.