Combining the effects of subwavelength resonators and very-high-permittivity substrates enables a high extent of miniaturization, even for very simple, split-loop resonators. Here, we demonstrate how requirements to the substrate's permittivity are connected with the desired extent of miniaturization and why materials with a relative permittivity of 30 <; ε <; 100, like lithium niobate, may offer a real possibility to miniaturize. For demonstration purposes, we designed, in line with this approach, an ultraminiature dual-band antenna to operate at 2.8 and 4.2 GHz. The antenna is fabricated using microfabrication techniques and studied experimentally. There is good agreement between the measurement and simulation results. The realized gain is about -5 dB for the first resonance, at which the size of the substrate-resonator block is λ/24. The obtained results demonstrate the potential of the suggested approach, which is expected to be applicable to a very wide class of subwavelength resonators and a wide variety of substrates with high permittivity.