Suppose a random sample of size $n$ is drawn from a normal population with mean $\mu$ and standard deviation $\sigma$ and that the sample has been censored either to the right or the left. Suppose the censoring is at a fixed point of the distribution or at a pre-specified sample percentage point, or is a combination of these two types of censoring. In this paper we present small sample bounded confidence intervals for $\mu$ and $\sigma$, based on a joint bounded confidence region at a confidence level with fixed bound. The limits for $\mu$ and $\sigma$ so obtained converge in probability, as $n \rightarrow \infty$, to the parameter values. The procedure of the paper allows similar results for some other scale-translation families of distributions. One such case, which is briefly discussed, is that of the exponential distribution with unknown initial point. The somewhat general applicability of the procedure mitigates the fact that it is not based on sufficient statistics.