BRACKETTVILLE, TX. BRACKETTVILLE, TEXAS. Brackettville, the county seat of Kinney County, is on U.S. Highway 90 twenty-two miles northeast of the Rio Grande and 125 miles west of San Antonio, near the geographic center of the county.RACKETTVILLE, TX. BRACKETTVILLE, TEXAS. Brackettville, the county seat of Kinney County, is on U.S. Highway 90 twenty-two miles northeast of the Rio Grande and 125 miles west of San Antonio, near the geographic center of the county.