Powerful Problems with the Precession of the Perihelion of Mercury
Another of the "proofs" of relativity is the General Relativity's prediction of the precession of Mercury's orbit.  Here, you'll find a quick explanation that there is a seeming difference between the Newtons classical prediction of this motion and what is actually happeneing.
Classical Prediction:5600 arc seconds per century
Actual:5557 arc seconds per century.
  
Einstein's equations in general relativity predict it perfectly and that is where most teachers end the story. The story is far from over at that point!
  
Firstly a man named Paul Gerber already solved the problem and published it seventeen years earlier than Einstein using classical Newtonian physics with the small adjustment that the propagation speed of gravity is not infinite.(1898) Einstein's equations were extraordinarily similar and it is very widely held that this is purely plagiary. However, Einstein named no previous sources in his publishing of general relativity, So one can believe that is was just an oversight instead of malicious intent but here is a good excerpt from a book that was written on the topic by a historian of science. Einstein: The Irrational Plagiarist
  
Here is a long story with math about Gerber. Here is more of the math explanation from another site once again pointing out that there is no need for relativity.
  
Additionally, the precession of binary stars do not match up to General relativity.
  
Another more recent theory concerns a more modern understanding of the solar magnetic field and the fact that it could very likely be the cause in the anomalous motion.