Relaxed notions of decidability widen the scope of automatic verification of hybrid systems. In quasi-decidability and $\delta$-decidability, the fundamental compromise is that if we are willing to accept a slight error in the algorithm's answer, or a slight restriction on the class of problems we verify, then it is possible to obtain practically useful answers. This paper explores the connections between relaxed decidability and the robust semantics of Metric Temporal Logic formulas. It establishes a formal equivalence between the robustness degree of MTL specifications, and the imprecision parameter $\delta$ used in $\delta$-decidability when it is used to verify MTL properties. We present an application of this result in the form of an algorithm that generates new constraints to the $\delta$-decision procedure from falsification runs, which speeds up the verification run. We then establish new conditions under which robust testing, based on the robust semantics of MTL, is in fact a quasi-semidecision procedure. These results allow us to delimit what is possible with fast, robustness-based methods, accelerate (near-)exhaustive verification, and further bridge the gap between verification and simulation.