  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.
