We propose a general framework for nonasymptotic covariance matrix estimation making use of concentration inequality-based confidence sets. We specify this framework for the estimation of large sparse covariance matrices through incorporation of past thresholding estimators with key emphasis on support recovery. This technique goes beyond past results for thresholding estimators by allowing for a wide range of distributional assumptions beyond merely sub-Gaussian tails. This methodology can furthermore be adapted to a wide range of other estimators and settings. The usage of nonasymptotic dimension-free confidence sets yields good theoretical performance. Through extensive simulations, it is demonstrated to have superior performance when compared with other such methods. In the context of support recovery, we are able to specify a false positive rate and optimize to maximize the true recoveries.