The effect of surface tension γ on cavity nucleation and growth in an incompressible neo-Hookean material of initial elasticity modulus E is examined. In a numerical analysis of potential energy limited to spherically symmetric configurations, a bifurcation problem is identified. If the applied uniform radial tensile dead-load is sufficiently large, a branch corresponding to cavity expansion bifurcates from the branch corresponding to cavity constriction due to surface tension. The bifurcation is interpreted in terms of a sudden growth to a finite cavity radius which bears some reminiscence of the snap-through buckling phenomenon in structural mechanics. The finding is compared to experimental results obtained in probe tack tests of soft adhesives where rapid cavity growth is also observed. Some of the experimental evidence is explained by assuming that in the adhesive there is a distribution of pre-existing cavities whose radius compares in a way to the length scale γ/E such as to make them susceptible to the influence of surface tension.