The concept of fuzzy metric space introduced by Kramosil and Michalek was later slightly modified by George and Veeramani who imposed three additional restrictions on it. A significant difference between these two concepts of fuzzy metrics is that fuzzy metric spaces in the sense of George and Veeramani do not admit completion, in general. This paper is devoted to go into detail on completable fuzzy metric spaces by means of the study of the impact on the completion of each one of the restrictions imposed by George and Veeramani in their definition of fuzzy metric. In this direction, we characterize those completable fuzzy metric spaces, in which just one of the three restrictions imposed by George and Veeramani is required. Various examples illustrate and justify the main results.