This study aims to address two central questions. First, are fine-tuned convolutional neural networks (CNNs) necessary for medical imaging applications? In response, we considered four medical vision tasks from three different medical imaging modalities, and studied the necessity of fine-tuned CNNs under varying amounts of training data. Second, to what extent the knowledge is to be transferred? In response, we proposed a layer-wise fine-tuning scheme to examine how the extent or depth of fine-tuning contributes to the success of knowledge transfer. Our experiments consistently showed that the use of a pre-trained CNN with adequate fine-tuning outperformed or, in the worst case, performed as well as a CNN trained from scratch. The performance gap widened when reduced training sets were used for training and fine-tuning. Our results further revealed that the required level of fine-tuning differed from one application to another, suggesting that neither shallow tuning nor deep tuning may be the optimal choice for a particular application. Layer-wise finetuning may offer a practical way to reach the best performance for the application at hand based on the amount of available data. We conclude that knowledge transfer from natural images is necessary and that the level of tuning should be chosen experimentally.