I'll use $\sqrt[k]{\cdot}$ to denote the principal real $k$-th root of a real-valued input, i.e. the maximum real $k$-th root if it exists and undefined otherwise.
Consider integers $n, x, y, z > 0$ such that $\sqrt[n]{x + \sqrt{y}} + \sqrt[n]{x - \sqrt{y}} = z$. Is it necessarily true that there exist integers $p, q > 0$ such that $\sqrt[n]{x + \sqrt{y}} = \frac{1}{2}(p + \sqrt{q})$? We could equivalently ask if (1) $\sqrt[n]{x + \sqrt{y}} - \sqrt[n]{x - \sqrt{y}}$ is the square root of an integer or (2) $4 \sqrt[n]{x^2 - y}$ is an integer.
It's trivially true for $n = 1$, and not hard to show for $n = 2$. Probably $n = 3$ is doable, but I'm sure there's a more general method. I also suspect we can say something similar when the condition is instead $\sqrt[n]{x + \sqrt{y}} - \sqrt[n]{x - \sqrt{y}} = z$. I'd be interested in knowing if these both fall into a more general condition with the same denesting result.
I know there are already many questions about denesting radicals on this site: here, here, and here, among others. I don't want to add to the noise, but I didn't see an answer to this question.
