Is it possible for integer square roots to add up to another?

I initially was wondering if it were possible for there to be three $x,y,z \in \mathbb{Q}$ and $\sqrt{x},\sqrt{y},\sqrt{z} \notin \mathbb{Q}$ such that $\sqrt{x} + \sqrt{y} = \sqrt{z}$. I had suspected not, but then I found $x = \dfrac{1}{2}, y = \dfrac{1}{2}$ and thus $\sqrt{x} + \sqrt{y} = \sqrt{2}$.

I suspect there are no integer solutions where the numbers are not all square, but I couldn't prove it. Nonetheless I figured I'd ask if it were possible when $x, y, z \in \mathbb{N}$ and $\sqrt{x},\sqrt{y},\sqrt{z} \notin \mathbb{N}$ that $\sqrt{x} + \sqrt{y} = \sqrt{z}$? And if not, can someone prove it?

If you square both sides, you have $x + y + 2\sqrt{xy} = z$ which can be satisfied if and only if $\sqrt{xy}$ is an integer (it can't be a half integer). So, that's the condition, if $xy$ is a square then you're fine, otherwise, its unsolvable.

Some examples for each variable $x$ and $y$ up to 100:

$\sqrt{x}+\sqrt{x}=2\sqrt{x}=\sqrt{4x}$. For example: $\sqrt{3}+\sqrt{3}=\sqrt{12}$.