Burn-in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, optimal burn-in procedures for a system with two types of failures (i.e., minor and catastrophic failures) are investigated. A new system surviving burn-in time b is put into field operation and the system is used under a warranty policy under which the manufacturer agrees to provide a replacement system for any system that fails to achieve a lifetime of at least w. Upper bounds for optimal burn-in time minimizing the total expected warranty cost are obtained under a more general assumption on the shape of the failure rate function which includes the bathtub shaped failure rate function as a special case.