In this paper we study a differential fault attack against ciphers having the same physical structure as in the Grain family. In particular we demonstrate our attack against Grain v1, Grain-128 and Grain-128a. The existing attacks by Berzati et al. (HOST 2009), Karmakar et al. (Africacrypt 2011) and Banik et al. (CHES 2012) assume a fault model that allows them to reproduce a fault at a particular register location more than once. However, we assume a realistic fault model in which the above assumption is no longer necessary, i.e., re-injecting the fault in the same location more than once is not required. In addition, towards a more practical framework, we also consider the situation in which more than one consecutive locations of the LFSR are flipped as result of a single fault injection.