The human genome can be segmented into topologically associating domains (TADs), which have been proposed to spatially sequester genes and regulatory elements through chromatin looping. Interactions between TADs have also been suggested, presumably because of variable boundary positions across individual cells. However, the nature, extent and consequence of these dynamic boundaries remain unclear. Here, we combine high-resolution imaging with Oligopaint technology to quantify the interaction frequencies across both weak and strong boundaries. We find that chromatin intermingling across population-defined boundaries is widespread but that the extent of permissibility is locus-specific. Cohesin depletion, which abolishes domain formation at the population level, does not induce ectopic interactions but instead reduces interactions across all boundaries tested. In contrast, WAPL or CTCF depletion increases inter-domain contacts in a cohesin-dependent manner. Reduced chromatin intermingling due to cohesin loss affects the topology and transcriptional bursting frequencies of genes near boundaries. We propose that cohesin occasionally bypasses boundaries to promote incorporation of boundary-proximal genes into neighboring domains.