Genome organization within the cell nucleus is a result of chromatin condensation achieved by histone tail-tail interactions and other nuclear proteins that counter the outward entropic pressure of the polymeric DNA. We probed the entropic swelling of chromatin driven by enzymatic disruption of these interactions in isolated mammalian cell nuclei. The large-scale decondensation of chromatin and the eventual rupture of the nuclear membrane and lamin network due to this entropic pressure were observed by fluorescence imaging. This swelling was accompanied by nuclear softening, an effect that we quantified by measuring the fluctuations of an optically trapped bead adhered onto the nucleus. We also measured the pressure at which the nuclear scaffold ruptured using an atomic force microscope cantilever. A simple theory based on a balance of forces in a swelling porous gel quantitatively explains the diffusive dynamics of swelling. Our experiments on decondensation of chromatin in nuclei suggest that its compaction is a critical parameter in controlling nuclear stability.