Josh:
Conveying information from overlapping features is always a challenge, whether it is points, lines or polygons. Some form of distribution process is typical for points and lines, like bus routes. LR is ideal for this because without redrawing anything or changing any actual geometry I can create side offsets of overlapping lines or points on my Centerlines through an offset distance value in another field. So I can assign different levels of side offset to create a distribution pattern that has a relationship to the date ranges of overlapping events. Because the geometry of the events is created at runtime in memory, it is very flexible, but the redraw time is somewhat slower than normal features, and a large number of features drawing at once will noticeably slow ArcMap down. However, once you have your event side offsets arranged you can export then to become actual features so they will display quickly.
Converting your existing overlapping lines to events is possible through the Locate Features Along Routes tool. You can save both datasets, since the features you currently have will be permanent to the original network, while the events are cartographically adjustable. Because changes can happen to the underlying network, the Roads and Highways extension is worth considering, because it helps create time stamped versions of your Centerline network as it changes over time and maintains event integrity in response to those changes according to the rules you set. The 10.2.2 version of the extension also lets you define segments without tracing using human readable relative limits, based on cross streets (or any other feature that crosses your road network, like City Boundaries) and offset distances along the road in a compass direction.
