You might find this interesting.
As seen from Earth the precession of Mercury's orbit is measured to be 5600 seconds of arc per century (one second of arc=1/3600 degrees). Newton's equations, taking into account all the effects from the other planets (as well as a very slight deformation of the sun due to its rotation) and the fact that the Earth is not an inertial frame of reference, predicts a precession of 5557 seconds of arc per century. There is a discrepancy of 43 seconds of arc per century.An error of 43 out of 5557 is nearly one part in 10^2. So I guess I'm not sure what you meant.
First my statement that Newton was accurate to 1 part in 107 came from Penrose's, The Road to Reality, pg. 390. And second your estimation of accuracy is not very accurate, it would be more accurate (I am fudging a little) if you took into account all the revolutions that Mercury makes in 100 years (apx. 415) and used that to provide a percentage (43/1,494,000= .0000287), not 1 part in 107 but closer.
The problem with that particular example is that because of the mass and proximity of the Sun to Mercury relativistic forces are in play and it was one of the examples that Einstein used to support the Theory of Relativity : )
An interesting web site on the differences and problems with the two theories is, http://www.coolissues.com/gravitation/gmetric.htm