Ternary Co oxides of the general formula RBaCo4O7+delta (where R = rare earth or Y) represent a family of materials that demonstrate impressive oxygen storage capabilities-as much as 3.25% by weight-at a relatively low temperature of similar to 350 degrees C. This behavior results from structural features amenable to formation of oxygen interstitials and the availability of wide channels offered by the structure for the facile diffusion and storage of oxygen. Remarkably, this material is also a model system for exploring geometrically frustrated magnetism due to the presence of interleaving Kagome and triangular cobalt sublattices. Unraveling the interplay of structure, oxygen stoichiometry, and phase behavior is important in optimizing the properties of this class of oxygen storage candidates and for broadening understanding of unusual magnetism. In this paper, we use thermogravimetric analysis, synchrotron X-ray diffraction, and neutron diffraction to explore how oxygen stoichiometry can be varied systematically and the impact of oxygen variability on phase behavior. In particular, we report the existence of a miscibility gap in lightly oxygenated YBaCo4O7+delta materials (0.0 < delta < 0.08) and the effects of the additional interstitial oxygen atoms on the material's nuclear structure and magnetism. Structural phase transitions that were previously observed in the pristine stoichiometric parent YBaCo4O7.0 material are suppressed by extra oxygen, leading to suppression of long-range magnetic order in favor of short-range correlations.