A method is developed for determining upper bounds for the quantiles of the distribution of the maximum of a fixed or random number of positive variates which are positively orthant dependent. (Such bounds are of practical interest e.g. as in setting the design codes of structures which must sustain random loads.) The bounds obtained using the method will be quite conservative unless the variates themselves are bounded. If they are bounded and the number of variates is fixed and large and/or the quantile desired is extremely high, the bound will approximate the true quantile of the Type 111 limiting distribution obtained by Gumbel [5]. Unlike Gumbel's asymptotic result, this method may be applied when the number of variates is small and so avoids some ambiguity about its applicability. On the other hand it is more complicated since it involves an application of a basic inequality for random variables.