While there are well-known demonstrations that children can use distributional information to acquire multiple components of language, the underpinnings of these achievements are unclear. In the current paper, we investigate the potential pre-requisites for a distributional learning model that can explain how children learn their first words. We review existing literature and then present the results of a series of computational simulations with Vector Space Models, a type of distributional semantic model used in Computational Linguistics, which we evaluate against vocabulary acquisition data from children. We focus on nouns and verbs, and we find that: (i) a model with flexibility to adjust for the frequency of events provides a better fit to the human data, (ii) the influence of context words is very local, especially for nouns, and (iii) words that share more contexts with other words are harder to learn.