Nearly all of the distribution-free tolerance intervals that have been considered in the statistical literature are such that each endpoint consists of a sample order statistic or is unbounded. For two-sided intervals, many other possibilities are available. One purpose of this paper is to call attention to these additional possibilities, and to discuss their advantages and disadvantages. The principal purpose, however, is to show how to construct specialized two-sided tolerance intervals for use in investigating situations of two types. For the first situation, the probability concentration in the central part of the population is investigated. For the other, it is desired to investigate the probability concentration about a specified point. In both situations, the specialized intervals presented are not difficult to apply. For the first situation, permissible additional information can be used to obtain intervals tending to have shorter lengths than those not exploiting this extra information. In the other situation, the relative emphasis placed on the part of the interval to one side of the specified point, as compared to the part on the other side, can be specified.