After watching quite a few episodes of CSI:Crime Scene Investigation I get the feeling that all crimes are solved. Is there any episodes where they actually fail in doing this?
Most of the crimes on CSI are wrapped up neatly by the end of a single episode. They also have longer story arcs that take anywhere from two or three episodes to an entire season to solve. The Blue Paint Killer, Miniature Killer, Dick and Jane Killer, and Dr. Jekyll story arcs are all examples of cases that take several episodes for the CSI team to solve. The pilot episode of the series also introduces a case that isn't solved until season two.
Very rarely there are cases that just go completely unsolved. One example is Sqweegel, which is based on a character from CSI creator Anthony Zuiker's series of Level 26 novels. Sqweegel was responsible for a series of attacks in one episode in season 11, but so far hasn't shown up again.
In the CSI: NY Season 4 episode "Down The Rabbit Hole", the killer - a professional assassin - manages to escape. While the team know the assassin is responsible, and so they technically solved the crime, their real name is never revealed and (AFAIK) they are never caught in any subsequent episode.