Summary This paper is in the line of the recent literature on weak instruments, which, following the seminal approach of Stock and Wright captures weak identification by drifting population moment conditions. In contrast with most of the existing literature, we do not specify a priori which parameters are strongly or weakly identified. We rather consider that weakness should be related specifically to instruments, or more generally to the moment conditions. In addition, we focus here on the case dubbed nearly-weak identification where the drifting DGP introduces a limit rank deficiency reached at a rate slower than root-T. This framework ensures the consistency of Generalized Method of Moments (GMM) estimators of all parameters, but at a rate possibly slower than usual. It also validates the GMM-LM test with standard formulas. We then propose a comparative study of the power of the LM test and its modified version, or K-test proposed by Kleibergen. Finally, after a well-suited rotation in the parameter space, we identify and estimate directions where root-T convergence is maintained. These results are all directly relevant for practical applications without requiring the knowledge or the estimation of the slower rate of convergence.