Visual grounding, localizing objects from natural language descriptions, represents a critical bridge between language and vision understanding. While multimodal large language models (MLLMs) achieve impressive scores on existing benchmarks, a fundamental question remains: can MLLMs truly visually ground with human-like sophistication, or are they merely pattern-matching on simplified datasets? Current benchmarks fail to capture real-world complexity where humans effortlessly navigate intricate references and recognize when grounding is impossible. To rigorously assess MLLMs' true capabilities, we introduce GroundingME, a benchmark that systematically challenges models across four critical dimensions: (1) Discriminative: distinguishing highly similar objects, (2) Spatial: understanding complex relational descriptions, (3) Limited: handling occlusions or tiny objects, and (4) Rejection: recognizing ungroundable queries. Through careful curation combining automated generation with human verification, we create 1,005 challenging examples mirroring real-world complexity. Evaluating 25 state-of-the-art MLLMs reveals a profound capability gap: the best model achieves only 45.1% accuracy, while most score 0% on rejection tasks. We explore two strategies for improvements: (1) test-time scaling selects optimal response by thinking trajectory to improve overall performance by up to 4.5%, and (2) data-mixture training boosts rejection accuracy from 0% to 27.9%. GroundingME thus serves as both a diagnostic tool revealing current limitations in MLLMs and a roadmap toward human-level visual grounding. Project page: https://groundingme.github.io