We present 3.7 arcsec (∼0.05 pc) resolution 3.2 mm dust continuum observations from the Institut de Radioastronomie Millimétrique Plateau de Bure Interferometer, with the aim of studying the structure and fragmentation of the filamentary infrared dark cloud (IRDC) G035.39–00.33. The continuum emission is segmented into a series of 13 quasi-regularly spaced (λobs ∼ 0.18 pc) 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 (a 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 emphasizes 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 (3.6 × 1023 cm−2 < N H, c < 8.0 × 1023 cm−2), mass (8.1 M⊙ < M c < 26.1 M⊙), and number density (6.1 × 105 cm−3 < n H, c, eq < 14.7 × 105 cm−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 ∼20–25 M⊙, 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 of 230 μG < B < 670 μG, the cores are susceptible to gravitational collapse. These results may imply a multilayered fragmentation process, which incorporates the formation of sub-filaments, embedded cores, and the possibility of further fragmentation.