OH(1720 MHz) masers are created by the interaction of supernova remnants with molecular clouds. These masers are pumped by collisions in warm, shocked molecular gas with OH column densities in the range 1016-1017, cm-2. Excitation calculations suggest that inversion of the 6049 MHz OH line may occur at the higher column densities that have been inferred from main-line absorption studies of supernova remnants with the Green Bank Telescope. OH(6049 MHz) masers have therefore been proposed as a complementary indicator of remnant-cloud interaction. This motivated searches for 6049 MHz maser emission from supernova remnants using the Parkes 63 m and Effelsberg 100 m telescopes, and the Australia Telescope Compact Array. A total of forty-one remnants have been examined by one or more of these surveys, but without success. To check the accuracy of the OH column densities inferred from the single-dish observations we modelled OH absorption at 1667 MHz observed with the Very Large Array towards three supernova remnants, IC 443, W44 and 3C 391. The results are mixed*the OH column is revised upwards in IC443, downwards in 3C391, and is somewhat reduced in W44. We conclude that OH columns exceeding 1017 cm-2 are indeed present in some supernova remnants and so the lack of any detections is not explained by low OH column density. We discuss the possibility that non-local line overlap is responsible for suppressing the inversion of the 6049 MHz line.