Can electrical noise cancel itself out?
You are correct about the equation's restrictions. The words you are looking for are correlated and uncorrelated noise. That equation is for valid uncorrelated noise sources only.

But it's a tall order to have all the noise sources in a circuit to be of the correlated variety, let alone be correlated in such that they match so perfectly and in exactly the right way so as to completely cancel each other out. It's kind of like hoping the random quantum motion of all particles in your starship just happen to all continuously be in the same direction so that you move forward with no fuel.

Indeed, correlation could lead to the noise being higher than that indicated by your equation.