High signal to noise ratio (SNR) consistency of model selection criteria in linear regression models has attracted a lot of attention recently. However, most of the existing literature on high SNR consistency deals with model order selection. Further, the limited literature available on the high SNR consistency of subset selection procedures (SSPs) is applicable to linear regression with full rank measurement matrices only. Hence, the performance of SSPs used in underdetermined linear models (a.k.a compressive sensing (CS) algorithms) at high SNR is largely unknown. This paper fills this gap by deriving necessary and sufficient conditions for the high SNR consistency of popular CS algorithms like $l_0$-minimization, basis pursuit de-noising or LASSO, orthogonal matching pursuit and Dantzig selector. Necessary conditions analytically establish the high SNR inconsistency of CS algorithms when used with the tuning parameters discussed in literature. Novel tuning parameters with SNR adaptations are developed using the sufficient conditions and the choice of SNR adaptations are discussed analytically using convergence rate analysis. CS algorithms with the proposed tuning parameters are numerically shown to be high SNR consistent and outperform existing tuning parameters in the moderate to high SNR regime.