This is absolutely false. RDJ has been acting since 1970 and hasn't had more than a 2 year gap between projects since 1983.
He may not have been a marquee name throughout that whole time but he had steady work throughout the drug problems and arrests. He was in 29 films between 1983 and 1996 including Chaplin, for which he was nominated for an Oscar.
And that's before considering the fact that his father was a fairly successful Actor/Writer/Director.
Downey was rich as hell at a very young age. That's the primary reason he had so much trouble with drugs. Something that plagued a lot of 80's era child actors that saw success (Corey Haim, Corey Feldman, Drew Barrymore, etc.. etc..).