Not necessarily true. Witness the picture below. USS Cabot, CVL-28, last surviving ship of the Independence-class light carriers from WWII. Sold to Spain, and bought back by a group that attempted to make it into a museum (like Intrepid and Yorktown). Money wouldn't come through, so it was sold for scrap, towed to Brownsville, TX and cut up.
Neat pic, thanks...Cabot was scrapped, along with hundreds of WW II ships, after the war. Oriskany, CV-34, was scuttled as an artificial reef in 2006.