We investigate numerically first a chaotic map interrupted by two small neighborhoods, each containing an attracting point, and secondly a periodically tilted box within which disorderly colliding disks can reach different attracting configurations, due to dissipation. For finite, arbitrarily small accuracy, both systems have basins of attraction that are indistinguishable from intermingled basins: any neighborhood of a point in phase space leading to one attractor contains points leading to the other attractor. A bifurcation destabilizing the fixed points or the disk configurations causes on-off intermittency; the disks then alternate between a "frozen" and a gaslike state.