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A joint Argonne National Laboratory (ANL)/Naval Research Laboratory (NRL) program is under way to investigate X-band dielectric-loaded accelerating (DLA) structures, using high-power 11.424-GHz radiation from the NRL Magnicon Facility. DLA structures offer the potential of a simple, inexpensive alternative to copper disk-loaded structures for use in high-gradient radio-frequency (RF) linear accelerators. A central purpose of our high-power test program is to find the RF breakdown limits of these structures. In this paper, we summarize the most recent tests results for two DLA structures loaded with different ceramics: alumina and (MCT). No RF breakdown has been observed up to 5 MW of drive power (equivalent to 8 MV/m accelerating gradient), but multipactor was observed to absorb a large fraction of the incident microwave power. The latest experimental results on suppression of multipactor using a TiN coating on the inner surface of the dielectric are reported. Although we did not observe dielectric breakdown in the structure, breakdown did occur at the ceramic joint, where the electric field is greatly enhanced. Lastly, the MCT structure showed significantly less multipactor for the same level RF field.