Confidence intervals for parameters that can be arbitrarily close to being unidentified are unbounded with positive probability (e.g. Dufour, Econometrica 65, pp. 1365-1387 and Pfanzagl, Journal of Statistical Planning and Inference, 75, pp. 9-20), and the asymptotic risks of their estimators are unbounded (Pötscher, Econometrica 70, pp. 1035-1065). We extend these “impossibility results” and show that all tests of size α concerning parameters that can be arbitrarily close to being unidentified have power that can be as small as α for any sample size even if the null and the alternative hypotheses are not adjacent. The results are proved for a very general framework that contains commonly used models. 1 Address for correspondence: Giovanni Forchini, Department of Econometrics and Business Statistics, Monash University, Clayton, Victoria 3800, Australia. E-mail: Giovanni.Forchini@BusEco.monash.edu.au 2 I thank Grant Hillier, Patrick Marsh, Don Poskitt and Richard Smith for useful and encouraging comments and discussions. I thank the editor John Stufken and a referee for very useful feedback. This research was partially supported by Australian Research Council grant DP0771445.