When two separate Bose-Einstein condensates are suddenly put into contact in a double-well potential, the resulting dynamics of the many-body wave function leads to a proliferation of peaks in the left-right number difference probability distribution. These peaks are Schrodinger cat states, but can also be interpreted as rainbows (fold caustics). In the mean-field (GPE) regime the peaks become singular and are places where the mean-field theory fails not just quantitatively but also qualitatively. Although it is well known that standard optical rainbows represent places where geometric optics fails and a wave theory must be used, the related phenomenon in atomic Josephson junctions correspond to the interesting situation where a classical field theory fails and must be replaced by a 2nd-quantized theory.