- We provide bounds for moments of sums of sequences of independent random vari- ables. Concentrating on uniformly bounded non-negative random variables, we are able to improve upon previous results due to Johnson, Schechtman and Zinn (1985) and Latala (1997). Our basic results provide bounds involving Stirling numbers of the second kind and Bell numbers. By deriving novel efiective bounds on Bell numbers and the related Bell function, we are able to translate our moment bounds to explicit ones, which are tighter than previous bounds. The study was motivated by a problem in operation research, in which it was required to estimate the Lp-moments of sums of uniformly bounded non-negative random variables (representing the processing times of jobs that were assigned to some machine) in terms of the expectation of their sum.