Snoring noise is generated by vibration of the soft tissues of the upper airway, principally those that form the back of the roof of the mouth (the soft palate) and its conical extension (the uvula). In addition to discord with bed partners, snorers are at much greater risk of obstructive sleep apnoea. This sleep-related breathing disorder is characterised by episodic upper airway obstruction with accompanying sleep disruption and consequent excessive daytime sleepiness, as well as an elevated risk of accidents and cardiovascular disease. The instability that leads to flow-induced oscillations characteristic of inspiratory snoring in the human upper airway may be modelled using a cantilevered flexible plate in a mean channel flow. However, the cantilever in existing models strictly only captures the dynamics of the uvula. In a more complete model, these dynamics will be augmented by their interaction with the motions of the soft palate-itself a flexible structure of higher effective stiffness-from which the uvula extends. To investigate how the elasticity of the soft palate affects uvula motion and their combined susceptibility to flow-induced oscillation a modification is made to the standard model. In a one-dimensional cantilevered flexible plate we allow thickness to vary as a function of length, thus effecting local changes in inertia and flexural rigidity. The overall cantilever therefore comprises a section representing the soft palate followed by a section of lower thickness that represents the uvula. The cantilever is attached to a rigid wall (hard palate) separating upper (nasal) and lower (oral) inlets of a rigid-walled channel (pharynx) conveying a viscous flow. This model is formulated using the open-source finite-element software library oomph-lib. A parametric study is performed in which the uvula-to-soft palate length and thickness ratios are varied whilst keeping their combined length constant. Results show that there is a critical uvula-length fraction that determines whether the uvula stabilizes or destabilises the system. A relatively 'short' uvula swings out of phase with the soft palate and these oscillations are observed to decay; the mode shapes involved are not predicted by a uniform-thickness plate model. By contrast, if the uvula is relatively 'long' the flexible plate motion is isolated to the uvular section and the oscillations grow in amplitude, indicating a net energy transfer from fluid to structure. Increasing the thickness, hence inertia and flexural rigidity, of a 'short' uvula, e.g., by oedema, makes the fluid-structure system more unstable. In this case if the oedema were aggrevated by the vibratory mechanical insult then it would be self-sustaining and imply a bidirectional relationship between snoring and oedema of the uvula. Anatomical variability is common in the lengths of the soft palate and uvula which may make some people more susceptible than others to uvulopalatal snoring. Palatal surgery for snoring has proved highly variable in its effectiveness. Modelling of palatal motion using this approach may help guide patient selection for and type of soft-palate surgery applied to treat this common and potentially disabling condition.