If you bypassed the high and low pressure control then it wouldn't matter if you were over charged or undercharged. The pressure controls are bypassed.
What was the AMP draw of the compressor. What was the compressor discharge line temperature? Discharge line temperature is a good indication of how hot the internal of the compressor is getting. This temperature should normally stay under 200°, but never over 220°. This sounds like it's tripping on the compressor thermal overload. The compressor cuts out and the condenser fan keeps running.
You left some important details out. What kind of motor is this? Semi-hermetic, hermetic, scroll, rotary, screw??? Is this a low, medium, or high temperature. Is it single phase or three phase. Does it have a current relay or a potential relay. Is it a PSC type compressor? There are a lot of factors to consider.
If you have an instructor and he never told you to put an amp clamp on that motor then you need to find a new instructor. Have you even checked the capacitors with a meter? Have you meg-ohmed the compressor windings? Have you checked to see if the compressor is shorted to ground? A tiny short can cause intermittent problems.
I know you are a student, but if you have to remember the details. You and your instructor are so worried about that refrigerant charge being the problem that you are overlooking the obvious electrical possibilities. If your refrigerant charge is even close you will not get trips on high and low pressure. If your pressure controls are jumped out then move on to the next option.
A COMPRESSOR CAN STILL RUN HOT EVEN IF THE HEAD PRESSURE IS NORMAL.
Faulty or improper size starting components can cause this. The pick up rating on the potential relay has to be correct or it might not open the start winding (if applicable). The compressor may keep running drawing high current, but may not high enough to trip your breaker. If your start relay contacts are sticking the first thing to check is the bleed resistor on the start caapacitor. If it is not present or not defective than the start capacitor wil discharge through the relay contacts as they close. The resistor allows the discharge to happen a lot faster and prevents arcing of the contacts. The contacts can get stuck or welded together.
This will eventually cause the motor to overheat and trip on the thermal overload too. Maybe you have it plugged into an oversized breaker.
If you are working on a PSC compressor then all that start component stuff is irrelevent. You still need to check the run capacitor for the proper MFD rating and voltage rating. Also make sure it's not shorted to the case.
Check these links for some simple help. When you get to the link right click to view them in full screen mode because they are othrwise hard to read. I have a lot more information I can share if you need help getting through school.
http://good-times.webshots.com/photo/280…
http://good-times.webshots.com/album/569…