Animal genomes are organized into topologically associated domains (TADs). TADs are thought to contribute to gene regulation by facilitating enhancer-promoter (E-P) contacts within a TAD and preventing these contacts across TAD borders. However, the absolute difference in contact frequency across TAD boundaries is usually less than 2-fold, even though disruptions of TAD borders can change gene expression by 10-fold. Existing models fail to explain this hypersensitive response. Here, we propose a futile cycle model of enhancer-mediated regulation that can exhibit hypersensitivity through bistability and hysteresis. Consistent with recent experiments, this regulation does not exhibit strong correlation between E-P contact and promoter activity, even though regulation occurs through contact. Through mathematical analysis and stochastic simulation, we show that this system can create an illusion of E-P biochemical specificity and explain the importance of weak TAD boundaries. It also offers a mechanism to reconcile apparently contradictory results from recent global TAD disruption with local TAD boundary deletion experiments. Together, these analyses advance our understanding of cis-regulatory contacts in controlling gene expression and suggest new experimental directions.