The Bruss-Robertson inequality gives a bound on the maximal number of elements of a random sample whose sum is less than a specified value, and the extension of that inequality which is given here neither requires the independence of the summands nor requires the equality of their marginal distributions. A review is also given of the applications of the Bruss-Robertson inequality, especially the applications to problems of combinatorial optimization such as the sequential knapsack problem and the sequential monotone subsequence selection problem.