We present reversible magnetization data of a high-quality Bi2Sr2CaCu2O8+delta single crystal and explore the occurrence of three-dimensional (3D)-xy critical behavior close to the bulk-transition temperature Tc and of Kosterlitz-Thouless (KT) behavior. Below and above the presumed Kosterlitz-Thouless transition temperature TKT we observe the characteristic two-dimensional (2D)-xy behavior: a downward shift in the crossing point phenomenon toward TKT as the field is decreased and sufficiently below TKT the characteristic 2D-xy relationship between the magnetization and the in-plane magnetic penetration depth lambdaab. In contrast, the measured temperature dependence of the superfluid density does not exhibit the characteristic KT behavior around the presumed TKT. The absence of this feature is traced back to the 2D- to 3D-xy crossover setting in around and above TKT. Invoking the Maxwell relation, the anomalous field dependence of the specific-heat peak is also traced back to the intermediate 2D-xy behavior. However, close to Tc we observe consistency with 3D-xy critical behavior in agreement with measurements of lambdaab.