Deep neural networks (DNNs) are being increasingly deployed as integral parts of software systems. However, due to the complex interconnections among hidden layers and massive hyperparameters, DNNs require being trained using a large number of labeled inputs, which calls for extensive human effort for collecting and labeling data. Spontaneously, to alleviate this growing demand, a surge of state-of-the-art studies comes up with different metrics to select a small yet informative dataset for the model training. These research works have demonstrated that DNN models can achieve competitive performance using a carefully selected small set of data. However, the literature lacks proper investigation of the limitations of data selection metrics, which is crucial to apply them in practice. In this paper, we fill this gap and conduct an extensive empirical study to explore the limits of selection metrics. Our study involves 15 selection metrics evaluated over 5 datasets (2 image classification tasks and 3 text classification tasks), 10 DNN architectures, and 20 labeling budgets (ratio of training data being labeled). Our findings reveal that, while selection metrics are usually effective in producing accurate models, they may induce a loss of model robustness (against adversarial examples) and resilience to compression. Overall, we demonstrate the existence of a trade-off between labeling effort and different model qualities. This paves the way for future research in devising selection metrics considering multiple quality criteria.