We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds Ca = Uη/γ of order unity, a near-cusp forms due to the contact line. The radius of curvature of the cusp’s tip scales by the slip length multiplied by an exponential of -Ca. The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We verify that modelling the air flow in the cusp region alone leads to predictions of the wetting failure in agreement with numerical simulations of the full air and liquid phases.