In this paper we introduce and study the concept of strong convergence in fuzzy metric spaces $(X,M,*)$ in the sense of George and Veeramani. This concept is related with the condition $\bigwedge_{t>0}M(x,y,t)>0$, which frequently is required or missing in this context. Among other results we characterize the class of $s$-fuzzy metrics by the strong convergence defined here and we solve partially the question of finding explicitly a \emph{compatible} metric with a given fuzzy metric.