The stream-based Max-Min diversification problem concerns the task of selecting a limited number of diverse instances from a data stream. The nature of the problem demands immediate and irrevocable decisions. The set-wise diversity to be maximized is the minimum distance among any pair of the selected instances. Standard algorithmic approaches for sequential selection disregard the possibility of selection failures, which is the situation where the last instances of the stream are picked by default to prevent having an incomplete selection. This defect can be catastrophic for the Max-Min diversification objective. In this paper we present the Failure Rate Minimization (FRM) algorithm that allows the selection of a set of disparate instances while reducing significantly the probability of having failures. This is achieved by means of both analytical and empirical techniques. FRM is put in comparison with relevant algorithms from the literature through simulations on real datasets, where we demonstrate its efficiency and low time complexity.