We demonstrate that certain supersymmetric Gödel-like universe solutions of supergravity are not solutions of string theory. This is achieved by realizing that supertubes are Bogomol'nyi-Prasad-Sommerfeld states in these spaces, and under certain conditions, when wrapping closed timelike curves, some world-volume modes develop negative kinetic terms. Since these universes are homogeneous, this instability takes place everywhere in space-time. We also construct a family of supergravity solutions which locally look like the Gödel universe inside a domain wall made out of supertubes, but have very different asymptotic structure. One can adjust the volume inside the domain wall so there will be no closed timelike curves, and then those spaces seem like perfectly good string backgrounds.