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In regression settings where explanatory variables have very low correlations and where there are relatively few effects each of large magnitude, it is commonly believed that the Lasso shall be able to find the important variables with few errors—if any. In contrast, this paper shows that this is not the case even when the design variables are stochastically independent. In a regime of linear sparsity, we demonstrate that true features and null features are always interspersed on the Lasso path, and that this phenomenon occurs no matter how strong the effect sizes are. We derive a sharp asymptotic trade-off between false and true positive rates or, equivalently, between measures of type I and type II errors along the Lasso path. This trade-off states that if we ever want to achieve a type II error (false negative rate) under a given threshold, then anywhere on the Lasso path the type I error (false positive rate) will need to exceed a given threshold so that we can never have both errors at a low level at the same time. Our analysis uses tools from approximate message passing (AMP) theory as well as novel elements to dea