One of the other core tenets of pragmatist epistemology is the pragmatic theory of truth, which is the claim that true statements are those which are useful-to-believe.
Consider that Bohr described language as more a matter of poetry than effective communication. The view that language is poetry useful for describing “images” and “mental connections” seems to have been influenced by the pragmatic theory of truth, which states that a theory is true if it is useful.
Consider also that what Bohr considers useful, is the description of “images” and “mental connections” between sense impressions. But, he does not assert that these are true, just useful.
The pragmatist tenet of metaphysics takes the pluralist view that “there is more than one sound way to conceptualize the world and its content.”
And this pragmatist tenet of metaphysics seems similar to Bohr’s idea of “complementarity”. After the Solvay Conference, Bohr was noted to have remarked:
“I can quite understand why we cannot speak about the content of religion in an objectifying language. The fact that different religions try to express this content in quite distinct spiritual forms is no real objection. Perhaps we ought to look upon these different forms as complementary descriptions which, though they exclude one another, are needed to convey the rich possibilities flowing from man’s relationship with the central order.” 
Here Bohr openly endorses mutually exclusive explanations. Or, to put it more plainly, he openly endorses contradictions. Not only that, he suggests that contradictions are required in order to make sense of religion!
He was known to attempt to apply complementarity to far more than religion, most notably to quantum physics, as we have mentioned earlier.
So, he certainly believed that quantum physics required one to embrace a contradiction, that something could be both a wave and a particle, for example.
So it does seem like Bohr ascribed to pragmatist metaphysics as well.
Pragmatist philosophy of science holds the view that:
“a scientific concept or theory should be evaluated by how effectively it explains and predicts phenomena, as opposed to how accurately it describes objective reality.”
Bohr certainly believed that a scientific theory should not be judged on how accurately it described reality. This can be readily inferred from his rejection of the fact that we can know physical reality and instead should confine ourselves to the study of appearances.
And he was more than happy to make predictions based on quantum theory, often in terms of probabilities. That was the entire point of the probabilities, to allow some kind of (non-casual) prediction. As long as one did not claim that they were accurately describing a physical and objective reality.
Therefore, we can say that Bohr was indeed influenced by pragmatism.
However, that is not his only major philosophical influence. As mentioned earlier, his philosophical mentor had positivist leanings.
Certain aspects of his philosophy, such as his focus on the linguistic and numerical aspects of science seem to suggest that they may have been influenced by positivism.
Whatever his specific philosophical influences, be they pragmatism, positivism and whatever else, what they have in common are heavy elements of Kantianism. To some extent, they reject reality as the proper starting place, reject the validity of the senses and attempt to reduce knowledge and science to a relationship between abstractions.
But, this is not how science is supposed to work.
Science is supposed to take reality as the starting point and conduct experiments in order to find out how reality works.
It is not supposed to reject the validity of the senses, it requires that the senses be valid in order to be able to know anything.
A proper philosophy of science does not treat science as a mere study of the relationships between abstractions.
It treats science as the study of reality.
It treats mathematics as the science of methods for identifying real relationships, not as a replacement for understanding said relationships.
Science is not supposed to reject objectivity, it requires it in order to function.
It is not supposed to reject the Law of Identity. It requires that to exist is to have a non-contradictory nature. That to be is to be something.
It is not supposed to reject causality, it requires that things are what they are and that without it nothing can be learned about reality. It requires that events have a cause.
What then are the consequences of all these bad philosophical influences? Quantum physics.
Which attempts to rob subatomic particles of their identity, assert actions without causes and which encourages people to ignore reality in favor of mathematics.
Quantum physics is strange and irrational not because the quantum world is really like that, but because of the philosophy of Bohr and others (such as Heisenberg and Schrodinger).
It need not be like that. A rational quantum physics can be developed. But, only if philosophies like that of Bohr are abandoned.
It is not too late…