We measure strong radio-frequency (RF) electric fields using rubidium Rydberg atoms prepared in a room-temperature vapor cell as field sensors. Electromagnetically induced transparency is employed as an optical readout. We RF-modulate the and Rydberg states with 50 and 100 MHz fields, respectively. For weak to moderate RF fields, the Rydberg levels become Stark-shifted, and sidebands appear at even multiples of the driving frequency. In high fields, the adjacent hydrogenic manifold begins to intersect the shifted levels, providing rich spectroscopic structure suitable for precision field measurements. A quantitative description of strong-field level modulation and mixing of S and D states with hydrogenic states is provided by Floquet theory. Additionally, we estimate the shielding of DC electric fields in the interior of the glass vapor cell.