When using the bootstrap in the presence of measurement error, we must first estimate the target distribution function; we cannot directly resample, since we do not have a sample from the target. These and other considerations motivate the development of estimators of distributions, and of related quantities such as moments and quantiles, in errors-in-variables settings. We show that such estimators have curious and unexpected properties. For example, if the distributions of the variable of interest, $W$, say, and of the observation error are both centered at zero, then the rate of convergence of an estimator of the distribution function of $W$ can be slower at the origin than away from the origin. This is an intrinsic characteristic of the problem, not a quirk of particular estimators; the property holds true for optimal estimators.