The growth of bicycling and bicycle network facilities in the United States warrants assessment of whether bicycle networks give populations safe access to valuable destinations, that is, a bicycle network must be sufficiently both safe and useful. The Bicycle Level of Traffic Stress metric is adapted to assign traffic stress values to street segments and intersections based on OpenStreetMap data, and cumulative job opportunity accessibility calculations are performed on the reduced, low-stress bicycle networks. Four metropolitan areas were analyzed: Minneapolis–St. Paul, MN; Miami, FL; Seattle, WA; and Washington, D.C. An “access gap” metric was proposed, comparing low-stress accessibility with higher-stress accessibility, as a way to gauge how much a city’s bicycle network could be improved through upgrading its higher-stress bicycle facilities. Accessibility was aggregated over a variety of geographic areas, including Core-Based Statistical Areas, municipal boundaries, and neighborhoods, demonstrating the applicability of the proposed analysis framework to a wide variety of scenarios. It was found that, generally, restricting bicycle travel to only low-stress networks results in universal reductions in accessibility, which vary in magnitude between metropolitan areas, and between neighborhoods within a given metropolitan area. Practical applications of a traffic stress-informed bicycle accessibility analysis framework are discussed.