In response to pathogens, the adaptive immune system generates specific antibodies that bind and neutralize foreign antigens. Understanding the composition of an individual's immune repertoire can provide insights into this process and reveal potential therapeutic antibodies. In this work, we explore the application of antibody-specific language models to aid understanding of immune repertoires. We introduce AntiBERTy, a language model trained on 558M natural antibody sequences. We find that within repertoires, our model clusters antibodies into trajectories resembling affinity maturation. Importantly, we show that models trained to predict highly redundant sequences under a multiple instance learning framework identify key binding residues in the process. With further development, the methods presented here will provide new insights into antigen binding from repertoire sequences alone.