The limiting effects of varying the thickness of a dielectric overlayer on planar double split-ring resonator (SRR) arrays are studied by terahertz time-domain spectroscopy. Uniform dielectric overlayers from 100 nm to 16 µm thick are deposited onto fixed SRR arrays in order to shift the resonance frequency of the electric response. We discuss the bounds of resonance shifting and emphasize the resulting limitations for SRR-based sensing. These results are presented in the context of typical biosensing situations and are compared to previous work and other existing sensing platforms.