Are there any disadvantages or complications to prenatal diagnosis?

Yes. Prenatal diagnosis of fetal conditions has both pros and cons - one con is that prenatal diagnosis is never 100% and can cause undue stress on parents during the remainder of the gestation. A desire to obtain prenatal diagnosis should only be done after careful thought about how you & your significant other will respond to the news and what you will do with the information.
Pre-natal diagnosis. There are risks to the obtaining of fetal tissue or amniotic fluid, but balanced against that are the risks of genetics abnormalities which are more common with pregnancies over 38-40. It is important to decide what course to take should a severe abnormality be discovered. If the decision is not to terminate the pregnancy regardless of the genetic outcome, it is probably better not to do either.