Axis off position

I'm having an issue with my setup keeping in position.  I noticed that when I move in slower G1 moves, everything seems to be ok.  I can setup an indicator, move off of it and down the lathe and then come back to that position and it holds on the indicator.  But when I introduced rapid moves, that went away. 

1. I did some research and decided to lower the Max_Velocity, Default_Accelertaion and Max_Acceleration.  This initially fixed my problem, although my motors are capable of much higher velocity and acceleration.  At this point, I could take long (15") moves down the lathe in a slow G1 move and then Rapid G0 back to the original position and it would hold every time.  I looped it so that it moved this way 40 times, and the indictor was never off position.
2.  I then started running actual programs, that included long and short moves, both rapid and slow.  Now we are seeing this issue again.  One time I might see the original position off by .054", and then I can reset it and run the same exact program and see the position off by .032" the next time.  It's always off, but it's never consistent with how much it is off.
3.  I go back to step 1 and do the long moves down and back and everything works as it should.

My guess is that it has something to do with the ramp up and ramp down of the acceleration, but I'm not sure where to start on fixing this issue.  Any help would be greatly appreciated.

LinuxCNC 2.7.15
Pico Systems PWM Servo Amplifier
Servo Motors are original to the machine. The machine is a LeBlond Regal 24 servo shift that originally was controlled with the GE Mark Century Tape Drive/ NC system. Motors were originally fitted with resolvers that I used with the Pico System resolver converters. I thought the resolvers might be what was causing the issue so I replaced them with encoders. Same problem, so obviously not the resolvers.
No latency warning from LinuxCNC.
Power supply is also original to the machine. Control Systems Research, Inc. Model 675, 60 VDC output.
Running the Pico System Universal PWM controller between the PC and the drives.

I assume the encoders are mounted on the motor. If you have the same problem with the encoders and resolvers, then look for mechanical loss in the couplings or screw mounts or bearings. If the loop is closed and following error is returning to zero, then it almost has to be mechanical.