Do individual dimensions in vector space have meaning?

Word2vec assigns an N-dimensional vector to given words (which can be considered a form of dimensionality reduction).

It turns out that, at least with a number of canonical examples, vector arithmetic seems to work intuitively. For example "king + woman - man = queen".

These terms are all N-dimensional vectors. Now, suppose, for simplicity, that $$N=3$$, $$\text{king} = [0, 1, 2], \text{woman} = [1, 1, 0], \text{man} = [2, 2, 2], \text{queen} = [-1, 0, 0]$$, then the expression above can be written as $$[0, 1, 2] + [1, 1, 0] - [2, 2, 2] = [-1, 0, 0]$$.

In this (contrived) example, the last dimension (king/man=2, queen/woman=0) suggests a semantic concept of gender. Aside from semantics, a given dimension could "mean" a part of speech, first letter, or really any feature or set of features that the algorithm might have latched onto. However, any perceived "meaning" of a single dimension might well just be a simple coincidence.

If we picked out only a single dimension, does that dimension itself convey some predictable or determinable information? Or is this purely a "random" artefact of the algorithm, with only the full N-dimensional vector distances mattering?