We provide finite sample bounds on the Normal approximation to the law of the least squares estimator of the projection parameters normalized by the sandwich-based standard errors. Our results hold in the increasing dimension setting and under minimal assumptions on the data generating distribution. In particular, we do not assume a linear regression function and only require the existence of finitely many moments for the response and the covariates. Furthermore, we construct confidence sets for the projection parameters in the form of hyper-rectangles and establish finite sample bounds on their coverage and accuracy. We derive analogous results for partial correlations among the entries of sub-Gaussian vectors. \end{abstract}