Can I sue a repair shop for work done in 2005?

In the US you may sue someone for just about anything.  In this case, you have a repair shop that did work for your pursuant to a contract to repair your bike.  in most states, the statute of limitations for a breach of contract claim is 6 years from the date of the breach.  the problem that you have is going to prove that the repair shop breached the agreement.  since 4+ years have passed there may be several reasons for the problems that you are having. t he repiar shop is going to deny that they did anything wrong.  Unless you have the paper work from the repair and you have an expert that will testify that the repair shop failed to do the correct reapair and that there is no other basis for having the problem with the bike, then you may have a good case.  Otherwise, you will have difficulty proving your claim.  I suggest meeting with a lawyer. most lawyers will initially meet with you for free.