In 1991 Neverov won the Capablanca Memorial in Havana, and was a winner of the Politiken Cup in 1994. He played for Ukraine in the 35th Chess Olympiad at Bled 2002. He took part in the FIDE World Chess Championship 2004 but was eliminated in the first round by Shakhriyar Mamedyarov. Neverov won the 2005/06 Hastings International Chess Congress. He tied for first with Merab Gagunashvili in the 2006/07 edition of the same event, and with Nidjat Mamedov and Vadim Malakhatko in 2007/08.