That's the curse of dimensionality in effect. In very high dimensions, the probability that two randomly chosen vectors are orthogonal increases, and conventional distance metrics become less useful since most vectors end up very far from each other.

In geometric terms, think of a circle embedded in a square. As the number of dimensions increases (e.g. a sphere embedded in a cube, a 4-dimensional ball embedded in a 4-dimensional cube, etc), most of the volume in the cube is outside the radius of the sphere. Any vectors you have effectively become sparse.

Basically this means that, while you need large dimensionality to model complex relations, high dimensionality makes it difficult to model these relationships (and need exponentially more data).

I don't really know why hundreds (I typically see 200-400) of dimensions are "better" than thousands, but there are a few justifications I can think of. You definitely get diminishing returns as you increase dimensionality, and size is a big concern. As an example, FastText's English word2vec is 300-d with 1 million word vectors, giving it a size of around 1 million * 300 floats * 4 bytes/float = 12GB (I haven't loaded it in a while so I forget if they do anything to compress it or if it gets bloated by something). Considering this often has to reside in-memory you really have to question whether marginal gains are worth a single model occupying on the order of 10x that much memory monetarily (and at sizes like that you even have on-disk storage, boot times, etc. as considerations)

Intuitively I can suggest the following: One of the problems is that even relatively huge corpuses like wikipedia can't deal with high dimensionality that well because even they likely lack the sheer size and diversity of content to "flesh it out", so to speak. My intuition tells me that this is due to the training process overfitting "clusters" of information together due to the size of the model. If you have too many dimensions, there's a lot of space for clusters to form, and with that will come some loss of semantic differentiability between clusters - or at least my hunch tells me so. You definitely want "computer" to be more associated with "mail" than "taupe" but if the frequencies of their associations are small they'll essentially be interpreted as noise or overfit. One thing to note is that word2vec embeddings are trained using a shallow neural net, and it's entirely possible for it to be a generic ML problem of too-many-parameters/bad network topology given the input when dimensions get too high.

With dimensions in this context it can be easy to forget that each additional dimension added can (potentially) add an order of complexity to the model - a smaller model lies on a hyperplane in the new vector space. What may happen is that an added dimension (by added, I mean before training, not after) might add some beneficial complexity for a specific subset of the model; e.g. if we were originally in very low dimensions, adding one dimension may allow king/queen and boy/girl to separate in the vector space based on gender( although in reality you can't create correspondences to individual dimensions and properties like this usually) but simply lead to noise or overfitting in other subsets. I think in very high dimensions this overfitting is likely to manifest itself in either too much clustering or strangely high similarity words resulting from outliers/noise in the data.

I've never really seen thousands of dimensions used in the wild, but I don't know of any papers that explicitly compare performance among different dimensions (word2vec is hard to evaluate with a single metric, though). Perhaps once you get to internal Google or Facebook levels of big data you could use distributed computing to make it work, but again, I haven't seen references to that.