Thermal fly-height control sliders are widely used in current hard disk drives to control and maintain subnanometer level clearance between the read-write head and the disk. The peculiar dynamics observed during touchdown/contact tests for certain slider designs is investigated through experiments and analytical modeling. Nonlinear systems theory is used to highlight slider instabilities arising from an unfavorable coupling of system vibration modes through an internal resonance condition, as well as the favorable suppression of instabilities through a jump condition. Excitation frequencies that may lead to large amplitude slider vibrations and the dominant frequencies at which slider response occurs are also predicted from theory. Using parameters representative of the slider used in experiments, the theoretically predicted frequencies are shown to be in excellent agreement with experimental results. This analytical study highlights some important air bearing surface design considerations that can help prevent slider instability as well as help mitigate unwanted slider vibrations, thereby ensuring the reliability of the head-disk interface at extremely low head-disk clearances.