Yes it can, this is because FXAA treats each frame individually as a flat 2D texture with no context as to what is what. So it just attempts to smooth edges that it can detect within set thresholds.
So while, it can break up these elements fairly well depending on the context.
It will not solve the temporal aliasing from the raw output(So in a sense it's not actually solving sub-pixel aliasing). Something that generally samples per fragment (like MSAA, sample based FSAA in some older games like FEAR , FEAR2, Condemned, and SGSSAA) when implemented correctly. Will to some degree or completely eliminate depending.
Introduce downsampling (Either via DSR, or regular driver downsampling. Which is OGSSAA) with FXAA and depending on the game, it will help out quite a bit with absolutely no downside to using FXAA in conjunction when the ratio factors of downsampling are high enough.
Ex scenario: http://screenshotcomparison.com/comparison/102090
Though this won't always solve a lot of temporal aliasing problems. Depending on the game it can do it a lot. Or not at all (Ex: Many UE3 games due to the extreme shader aliasing and harsh specular aliasing)
The same also applies to SMAA 1x. And depending on the scenario one can objectively work better in certain ways.