In this paper, a robust inventory problem with uncertain cumulative demands is considered. Interval-budgeted uncertainty sets are used to model possible cumulative demand scenarios. It is shown that under discrete budgeted uncertainty the robust min–max problem can be solved in polynomial time. On the other hand, for continuous budgeted uncertainty, the problem is weakly NP-hard. It can be solved in pseudopolynomial time and admits an FPTAS for nonoverlapping cumulative demand intervals.