The paradoxes are because the theory is formulated in terms of complicated mathematical expressions that do not map neatly onto a naive understanding of macroscopic reality.

The entire problem of learning QM is, that one has to learn not to trust ones intuition and instead look at what the math is showing. For example the supposed paradox of being sometimes a wave and sometimes a particle is only there if you insist to use the macroscopic analogies of wave and particle. In the full theory, the object is always a vector in an Hillbertspace, which is a concept that does not map neatly onto some combination of wave and particle.

In what circumstance does that not describe a paradox?

For instance, Russel's paradox arises from an inconsistent naive set theory.

Ah, good example. I guess that's fair.