is this fixed speed or is the drive system equipped with frequency converter?
what is the power rating? onshore/offshore? is this a high pressure service? type of compression service (hydrogen, refrigerant, etc) and is gas dirty subjecting machine to fouling, polymerization, etc.? any performance degradation noticed or changes recently by the process control? any insight on the history of the machine? is it drive through casing? could you post a snapshot of the vibration log report? any recent start up/shutdown? maintenance/overhaul? is this a heavy rotor (subject to rotor bow on cold stand by) ?

Just a few questions that came to mind...

By the way, compressor casing is normally stiff so it is normal practice to disregard its impact on rotor-dynamic behavior (i.e. infinite support stiffness).

----
The customer did add a weight to the balance line for some reason. No one (including customer) seems to know why. History of the machine is bit foggy and since this extra vibration never caused an alarm and we are in the middle of an overhaul, I'm trying to keep my research to our side only. I'm trying not include the customer at this point. We're too far along to make any real changes anyway.

I'm here to satiate my curiosity, if I can.

Out of curiosity, do you have naphta injection lines installed on each impeller stages ? Coker gas compressor are notorious for causing deposit through out impellers. The swing of MW can also be important and these machines are often required to provide wide operating envelop flexibility (turndown) with limited process control method(s) (suction throttling and recycling). I also expect this to be a back to back machine, otherwise please advise. The running speed looks descent, but maybe it is slightly pushed and on the high side. All this is of course very general.
I think a thorough examination of the lateral analysis report would give an idea whether the machine has been pushed rotordynamically. Which does not mean it is not OK, but that will say where it initially stands and how much room do exist when performance deteriorates or when there is an upset somewhere. For instance you could examine damped response analysis (separation margins, the amplification factors) and the shape of the vibration amplitude nears the picks. If you see a double pick, may want to check also if there is no critically damped speed between the critical speeds and the operating speed. Again this is a good exercise to understand the machine design. Think the other key info is vibration log reports.

You indicated there was no trip/alarm so everything is withing the OEM design settings - why worry then?
Are the vibration levels still in the allowance / limits stated in the API617 - Ch 1 & 2?