This paper presents four experiments on the syntactic structure of fragments, i.e. nonsentential utterances with propositional meaning and illocutionary force (Morgan, 1973). The experiments evaluate the predictions of two competing theories of fragments: Merchant's (2004) movement and deletion account and Barton & Progovac's (2005) nonsentential account. Experiment 1 provides evidence for case connectivity effects, this suggests that there is indeed unarticulated linguistic structure in fragments (unlike argued by Barton & Progovac 2005). Experiments 2-4 address a central prediction of the movement and deletion account: only those constituents which may appear in the left periphery are possible fragments. Merchant et al. (2013) present two...