We study the structure and stability of closed-ring carbon nanotubes using a theoretical model based on the Brenner-Tersoff potential. Many metastable structures can be produced. We focus on two methods of generating such structures. In the first, a ring is formed by geometric folding and is then relaxed into minimum energy using a minimizing algorithm. Short tubes do not stay closed. Yet tubes longer than 18 nm are kinetically stable. The other method starts from a straight carbon nanotube and folds it adiabatically into a closed-ring structure. The two methods give strikingly different structures. The structures of the second method are more stable and exhibit two buckles, independent of the nanotube length. This result is in strict contradiction to an elastic shell model. We analyze the results for the failure of the elastic model.