Quasiopenness seems to be generic to multi-field models of single-bubble open inflation. Instead of producing infinite open universes, these models actually produce an ensemble of very large but finite inflating islands. In this paper we study the possible constraints from CMB anisotropies on existing models of open inflation. The effect of supercurvature anisotropies combined with the quasiopenness of the inflating regions make some models incompatible with observations, and severely reduces the parameter space of others. Supernatural open inflation and the uncoupled two-field model seem to be ruled out due to these constraints for values of $\Omega_0\lesssim0.98$. Others, such as the open hybrid inflation model with suitable parameters for the slow roll potential can be made compatible with observations.