Owing to its intrinsically lubricious property, graphene has a high potential to be an atomically thin solid lubricant for sliding interfaces. Despite its ultrahigh breaking strength at the nanoscale, graphene often fails to maintain its integrity when subjected to macroscale tribological tests. To reveal the true wear characteristics of graphene, a nanoscale diamond tip was used to scratch monolayer graphene mechanically exfoliated to SiO2 substrates. Our experimental results show that while graphene exhibited extraordinary wear resistance in the interior region, it could be easily damaged at the step edge under a much lower normal load (similar to 2 orders of magnitude smaller). Similar behavior with substantially reduced wear resistance at the edge was also observed for monatomic graphene layer on graphite surface. Using molecular dynamics simulations, we attributed this markedly weak wear resistance at the step edge to two primary mechanisms, i.e, atom-by-atom adhesive wear and peel. induced rupture. Our findings shed light on the paradox that graphene is nanoscopically strong yet macroscopically weak. As step edge is ubiquitous for two-dimensional materials at the macroscale, our study also provides a guiding direction for maximizing the mechanical and tribological performance of these atomically thin materials.