We present 3.7 arcsec (~0.05 pc) resolution 3.2 mm dust continuum observations from the IRAM PdBI, with the aim of studying the structure and fragmentation of the filamentary Infrared Dark Cloud G035.39-00.33. The continuum emission is segmented into a series of 13 quasi-regularly spaced (~0.18pc) cores, following the major axis of the IRDC. We compare the spatial distribution of the cores with that predicted by theoretical work describing the fragmentation of hydrodynamic fluid cylinders, finding a significant (factor of ~8) discrepancy between the two. Our observations are consistent with the picture emerging from kinematic studies of molecular clouds suggesting that the cores are harboured within a complex network of independent sub-filaments. This result emphasises the importance of considering the underlying physical structure, and potentially, dynamically important magnetic fields, in any fragmentation analysis. The identified cores exhibit a range in (peak) beam-averaged column density (3.6x1023cm−2<NH,c<8.0x1023cm−2), mass (8.1M⊙<Mc<26.1M⊙), and number density (6.1x105cm−3<nH,c,eq<14.7x105cm−3). Two of these cores, dark in the mid-infrared, centrally-concentrated, monolithic (with no traceable substructure at our PdBI resolution), and with estimated masses of the order ~20-25M⊙, are good candidates for the progenitors of intermediate-to-high-mass stars. Virial parameters span a range 0.2<αvir<1.3. Without additional support, possibly from dynamically important magnetic fields with strengths of the order 230μG<B<670μG, the cores are susceptible to gravitational collapse. These results may imply a multi-layered fragmentation process, which incorporates the formation of sub-filaments, embedded cores, and the possibility of further fragmentation