In order to study the fragmentation of massive dense cores we observed the 1.3 mm continuum emission of four massive cores with the Plateau de Bure Interferometer in the most extended configuration. The broad-band correlator units also revealed emission from complex organic molecules, which was resolved down to ~300-700 AU and could be tracing disk structures. Concerning the continuum emission, we detected dust condensations down to ~0.3 Msun and separate millimeter sources down to 0.4'' or ~1000 AU, comparable to the sensitivities and separations reached in optical/infrared studies of clusters. This, in combination with additional cores from the literature observed at similar mass sensitivity and spatial resolution, allowed us to build a sample of 19 protoclusters with luminosities spanning three orders of magnitude. Among the 19 regions, 30% show no signs of fragmentation, while 50% split up into >~ 4 millimeter sources. We compiled a list of properties for the 19 massive dense cores, such as bolometric luminosity, total mass, and evolutionary stage indicators, and found no correlation of any of these parameters with the fragmentation level. Radial intensity profiles of single-dish submillimeter emission and Spectral Energy Distributions of the 19 massive dense cores were fitted simultaneously with a spherical core model assuming that the density and temperature decrease with radius following power-laws, and we studied the relation between density structure of the dense cores and their fragmentation level. We find a weak (inverse) trend of fragmentation level and density power-law index, with steeper density profiles tending to show lower fragmentation, and vice versa. In addition, we find a clear trend of fragmentation increasing with density within a given radius, which is consistent with Jeans fragmentation. Finally, in order to investigate the combined effects of the magnetic field, radiative feedback, and turbulence in the fragmentation process, we compared our observations to radiation magnetohydrodynamic simulations and found that the low-fragmented regions are reproduced well in the strongly magnetized core case, which tend to yield more concentrated density profiles than the weakly magnetized case (Palau et al. 2013, ApJ, 762, 120; Palau et al. 2014, submitted).Read less <