We propose a method for accurately detecting bioacoustic sound events that is robust to overlapping events, a common issue in domains such as ethology, ecology and conservation. While standard methods employ a frame-based, multi-label approach, we introduce an onset-based detection method which we name Voxaboxen. It takes inspiration from object detection methods in computer vision, but simultaneously takes advantage of recent advances in self-supervised audio encoders. For each time window, Voxaboxen predicts whether it contains the start of a vocalization and how long the vocalization is. It also does the same in reverse, predicting whether each window contains the end of a vocalization, and how long ago it started. The two resulting sets of bounding boxes are then fused using a graph-matching algorithm. We also release a new dataset designed to measure performance on detecting overlapping vocalizations. This consists of recordings of zebra finches annotated with temporally-strong labels and showing frequent overlaps. We test Voxaboxen on seven existing data sets and on our new data set. We compare Voxaboxen to natural baselines and existing sound event detection methods and demonstrate SotA results. Further experiments show that improvements are robust to frequent vocalization overlap.