Phenomena in diverse contexts such as wetting, biological assembly, and manufacturing are attributed to the three-phase line tension. However, decades of line tension estimates based on contact angles of droplets controversially span 6 orders of magnitude, raising the question of which measurements are authoritative. Here, we show with experiments and calculations that contact angles fail to estimate line tension regardless of length scale, technique, and measurement quality. Line tension measurements based on contact angles are driven by two distinct and spurious mechanisms: body forces under ideal conditions, and data scatter under noisy conditions.