Given a sample of data, it is always possible to construct non-parametric (i.e. confidence intervals which do not assume that the data are related to any particular probability distribution) confidence intervals about properties such as the median and the range of the data. So in the sense of there being CI's for some useful properties of a data sample, these always exist.
However, in the sense of there being CI's for specific properties assuming particular distributions, they may not exist. For example, if a series of data points are generated from a Cauchy distribution, calculating the mean will provide little or no useful information on the parameters of the distribution because the Cauchy distribution's mean is undefined, as is its variance. Hence, you can't calculate the confidence interval for the mean or variance for a sample of random data generated by the Cauchy distribution because they don't exist. The reason that the mean, for example, doesn't exist for the Cauchy distribution is that it is mathematically undefined. This answer
Essentially if there is a parameter which exists and can be estimated with real numbers, it will be possible to calculate a confidence interval, at least by bootstrapping if all else fails.