Initial condition estimates and model representations of clouds, precipitation, relative humidity, ice, and aerosols are a major source of weather and climate prediction error. Forecasts and observations of these variables often result in values that are just a few error standard deviations from their bounds. For example, ensemble precipitation forecasts can produce distributions in which the mean is roughly one ensemble standard deviation bigger than precipitation's lower bound of zero. In such cases, distributions tend to be highly skewed, and poorly approximated by Gaussian distributions. Remote sensors of cloud/rain typically yield observations that are non-linear functions of the model variables used to predict them. Here, we present new methodologies that enable the Local Ensemble Transform Kalman Filter (LETKF) to better approximate a fully non-linear Bayesian data assimilation scheme for observations of these variables. The method is based around a deterministic ensemble Kalman filter that, unlike the LETKF, assimilates observations one after the other. However, we have discovered how to express the results in terms of a symmetric transformation of the original ensemble perturbations. Such symmetric transformations are important to the LETKF approach because they maximize the spatial continuity of analysis perturbations. The approach allows the standard LETKF assumption of a Gaussian prior and a Gaussian observation likelihood to be replaced by, for example, gamma prior and inverse-gamma observation likelihood for variables bounded on one side and a Beta prior and Binomial observation likelihood for variables bounded on two sides. When the observation operator is non-linear, the LETKF often produces an analysis ensemble of the observed variable that is inconsistent with the underlying model variables. Our new approach eliminates this inconsistency with a Gauss-Newton iteration. Localization is achieved through ensemble "squeezing". In idealized data assimilation experiments, the Bounded Variable Ensemble Transform (BVET) enhanced LETKF profoundly out-performs the unenhanced LETKF.