Despite a great deal of recent work on the finite-sample properties of estimators and tests for linear regressions with a single endogenous regressor and weak instruments, there has been much less work on testing overidentifying restrictions in these circumstances. We study various asymptotic tests for overidentification in models estimated by instrumental variables (two-stage least squares), and by limited-information maximum likelihood. We show that all the statistics used by these tests are, like the statistics used for inference on the regression coefficient, functions of only six quadratic forms in the two endogenous variables of the model. They are closely related to the well-known test statistic of Anderson and Rubin. We show that the distributions of the overidentification statistics have an ill-defined limit as the strength of the instruments tends to zero along with a parameter related to the correlation between the disturbances of the two equations of the model. Simulation experiments demonstrate that this means that it is impossible to perform reliable inference near the point at which the limit is ill-defined. Some bootstrap procedures are proposed which alleviate the problem, and allow reliable inference when the instruments are not too weak.