This paper presents a study of the breakdown of the Navier–Stokes equations in hypersonic viscous flows over a sharp cone tip and a hollow cylinder/flare geometry. Investigations are performed through detailed comparisons of the numerical results obtained with continuum and particle techniques. The objective of the study is to predict conditions under which the continuum approach may be expected to fail. A modified breakdown parameter is proposed that can predict the failure of the continuum approach accurately for the simple cone flow and fairly well for the more complex cylinder/flare flow. The study of continuum breakdown is the first step toward development of a hybrid numerical code.