Intelligent systems are transforming the world, as well as our healthcare system. We propose a deep learning-based cough sound classification model that can distinguish between children with healthy versus pathological coughs such as asthma, upper respiratory tract infection (URTI), and lower respiratory tract infection (LRTI). To train a deep neural network model, we collected a new dataset of cough sounds, labelled with a clinician’s diagnosis. The chosen model is a bidirectional long–short-term memory network (BiLSTM) based on Mel-Frequency Cepstral Coefficients (MFCCs) features. The resulting trained model when trained for classifying two classes of coughs—healthy or pathology (in general or belonging to a specific respiratory pathology)—reaches accuracy exceeding 84% when classifying the cough to the label provided by the physicians’ diagnosis. To classify the subject’s respiratory pathology condition, results of multiple cough epochs per subject were combined. The resulting prediction accuracy exceeds 91% for all three respiratory pathologies. However, when the model is trained to classify and discriminate among four classes of coughs, overall accuracy dropped: one class of pathological coughs is often misclassified as the other. However, if one considers the healthy cough classified as healthy and pathological cough classified to have some kind of pathology, then the overall accuracy of the four-class model is above 84%. A longitudinal study of MFCC feature space when comparing pathological and recovered coughs collected from the same subjects revealed the fact that pathological coughs, irrespective of the underlying conditions, occupy the same feature space making it harder to differentiate only using MFCC features.