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Inflation of hollow elastic structures can become unstable and exhibit a runaway phenomenon if the tension in their walls does not rise rapidly enough with increasing volume. Biological systems avoid such inflation instability for reasons that remain poorly understood. This is best exemplified by the lung, which inflates over its functional volume range without instability. The goal of this study was to determine how the constituents of lung parenchyma determine tissue stresses that protect alveoli from instability-related overdistension during inflation. We present an analytical model of a thick-walled alveolus composed of wavy elastic fibres, and investigate its pressure–volume behaviour under large deformations. Using second-harmonic generation imaging, we found that collagen waviness follows a beta distribution. Using this distribution to fit human pressure–volume curves, we estimated collagen and elastin effective stiffnesses to be 1247 kPa and 18.3 kPa, respectively. Furthermore, we demonstrate that linearly elastic but wavy collagen fibres are sufficient to achieve inflation stability within the physiological pressure range if the alveolar thickness-to-radius ratio is greater than 0.05. Our model thus identifies the constraints on alveolar geometry and collagen waviness required for inflation stability and provides a multiscale link between alveolar pressure and stresses on fibres in healthy and diseased lungs.