We present one way in which combinatorial designs can be used to give conditionally perfect secret sharing schemes. Schemes formed in this way have the advantage over classical secret sharing schemes of being easily adapted for use as compartmentalized or hierarchical access structures. We study the problem of completion of structures, given partial information, to obtain measures of how closely the behaviour of the secret sharing schemes approaches to ideal behaviour in practice. It may happen that part of a combinatorial design can never be reconstructed from a subset of a minimal defining set. That is, to find the blocks of what is called the strongbox of a given minimal defining set of a design, we must have the whole of the minimal defining set and be able to complete the whole design. The strongbox is that part of the design which may most safely be used to hold secret information. We study the size of the strongbox.