In this paper we consider a class of partially adaptive one-step M-estimators for the non-linear regression model with dependent observations. Those estimators adapt themselves with respect to a measure of the tailthickness of the disturbance distribution (as well as to a measure of the scale). The large-sample behavior of those estimators is examined theoretically for general disturbance distributions and numerically for various specific ones. The estimators considered are motivated by the Student-t maximum-likelihood estimator. Given appropriate specifications of the adaptation parameter the estimators are asymptotically efficient on the family of Student-t distributions including the normal distribution.