Summary: A covering system is a system of $k$ arithmetic progressions whose union includes all integers. It is a disjoint covering system (or exact covering system) if the progressions are also pairwise disjoint, so that each integer is covered exactly once. This paper presents upper bounds on the number of consecutive integers which need to be checked to determine whether a covering system is a disjoint covering system. The bounds depend only on the number of congruences in the system. The results provide an analog of a theorem by R. B. Crittenden and C. L. Vanden Eynden from 1969 and are presented as solutions to some variations of a 2002 AIME Problem about painting a picket fence.