Neural networks are being used to make new types of empirical chemical models as inexpensive as force fields, but with accuracy similar to the ab initio methods used to build them. In this work, we present a neural network that predicts the energies of molecules as a sum of intrinsic bond energies. The network learns the total energies of the popular GDB9 database to a competitive MAE of 0.94 kcal/mol on molecules outside of its training set, is naturally linearly scaling, and applicable to molecules consisting of thousands of bonds. More importantly, it gives chemical insight into the relative strengths of bonds as a function of their molecular environment, despite only being trained on total energy information. We show that the network makes predictions of relative bond strengths in good agreement with measured trends and human predictions. A Bonds-in-Molecules Neural Network (BIM-NN) learns heuristic relative bond strengths like expert synthetic chemists, and compares well with ab initio bond order measures such as NBO analysis.