How crucial are power-of-two textures?

My understanding is that only really old GPUs require textures to have power-of-two dimensions, or is this still necessary for some mobile devices or with mipmapping? I really need to use the actual size of certain textures without forcing them to a power of two, unless this will create compatibility problems.
Also, I've noticed that at least in Unity 5.0, disabling power-of-two conversion has no effect on the texture when actually displayed via a material (it looks different in the inspector window but not when displayed on a quad), so I assume that it's still used by default regardless of the texture's settings?

 

There is a difference in optimization. Check your texture size depending on this parameter.

It may not make a difference on a simple project because your computer is so powerful that it doesn't care about bad optimisation on that scale but you can't ignore that on a commercial product.