Complex dynamical systems often have tipping points and exhibit catastrophic regime shift. Despite the notorious difficulty of predicting such transitions, accumulating studies have suggested the existence of generic early-warning signals (EWSs) preceding upcoming transitions. However, previous theories and models were based on the effect of the intrinsic noise (IN) when a system is approaching a critical point, and did not consider the pervasive environmental fluctuations or the extrinsic noise (EN). Here, we extend previous theory to investigate how the interplay of EN and IN affects EWSs. Stochastic simulations of model systems subject to both IN and EN have verified our theory and demonstrated that EN can dramatically alter and diminish the EWS. This effect is stronger with increasing amplitude and correlation time scale of the EN. In the presence of EN, the EWS can fail to predict or even give a false alarm of critical transitions.