Successful deployment of Deep Neural Networks (DNNs) requires their validation with an adequate test set to ensure a sufficient degree of confidence in test outcomes. Although well-established test adequacy assessment techniques have been proposed for DNNs, we still need to investigate their application within a comprehensive methodology for accurately predicting the fault detection ability of test sets and thus assessing their adequacy. In this paper, we propose and evaluate TEASMA, a comprehensive and practical methodology designed to accurately assess the adequacy of test sets for DNNs. In practice, TEASMA allows engineers to decide whether they can trust high-accuracy test results and thus validate the DNN before its deployment. Based on a DNN model's training set, TEASMA provides a procedure to build accurate DNN-specific prediction models of the Fault Detection Rate (FDR) of a test set using an existing adequacy metric, thus enabling its assessment. We evaluated TEASMA with four state-of-the-art test adequacy metrics: Distance-based Surprise Coverage (DSC), Likelihood-based Surprise Coverage (LSC), Input Distribution Coverage (IDC), and Mutation Score (MS). Our extensive empirical evaluation across multiple DNN models and input sets such as ImageNet, reveals a strong linear correlation between the predicted and actual FDR values derived from MS, DSC, and IDC, with minimum R^2 values of 0.94 for MS and 0.90 for DSC and IDC. Furthermore, a low average Root Mean Square Error (RMSE) of 9% between actual and predicted FDR values across all subjects, when relying on regression analysis and MS, demonstrates the latter's superior accuracy when compared to DSC and IDC, with RMSE values of 0.17 and 0.18, respectively. Overall, these results suggest that TEASMA provides a reliable basis for confidently deciding whether to trust test results for DNN models.