The remineralization depth of particulate organic carbon (POC) fluxes exported from the surface ocean exerts a major control over atmospheric CO₂ levels. According to a long‐held paradigm most of the POC exported to depth is associated with large particles. However, recent lines of evidence suggest that slow‐sinking POC (SSPOC) may be an important contributor to this flux. Here we assess the circumstances under which this occurs. Our study uses samples collected using the Marine Snow Catcher throughout the Atlantic Ocean, from high latitudes to midlatitudes. We find median SSPOC concentrations of 5.5 μg L−1, 13 times smaller than suspended POC concentrations and 75 times higher than median fast‐sinking POC (FSPOC) concentrations (0.07 μg L−1). Export fluxes of SSPOC generally exceed FSPOC flux, with the exception being during a spring bloom sampled in the Southern Ocean. In the Southern Ocean SSPOC fluxes often increase with depth relative to FSPOC flux, likely due to midwater fragmentation of FSPOC, a process which may contribute to shallow mineralization of POC and hence to reduced carbon storage. Biogeochemical models do not generally reproduce this behavior, meaning that they likely overestimate long‐term ocean carbon storage. Key Points Suspended, slow‐, and fast‐sinking particulate organic carbon (POC) contributes 94:6:<1% to total POC concentration, respectively POC flux below the mixed layer is often dominated by slow‐sinking particles In situ generation of slow‐sinking POC below the mixed layer, via midwater fragmentation, may lead to shallower mineralization of particles