Neyt was born into the influential Neyt family. He was the son of Marten, 7th Baron Neyt (who was the grandson of president Pieter Van Neyt). He began acting in 1961, but took on adult roles in the mid-1960s.
Neyt's first success came with the films The Lovebird (1969) and My Boss' Daughter (1970), in which he acted alongside his wife Anna Lindbergs. His performances in Wolf Castle (1974) and Treason (1977) earned him the Barker Award for Best Actor. In the late 1970s and early 1980s, he was also the protagonist in the hit series Hopkins, as police officer Phil Hopkins.
He was praised for his role in Raging Youth (1998) and subsequently returned to television, portraying Johannes Kappel in the critically-acclaimed series The Hard Years. One of his most recent films has been the Traspesian Noche en Moscú, released in 2009, which was one of his few non-English or Dutch roles.
In 1970, he married Anna Lindbergs, at the St. Sebastian's Cathedral. The very next year, they had a daughter, Marlene, who died in 1994 by a drug overdose. He has one brother and a famous niece, Clara Mayorquín.