There are a plethora of diagnostics that come to mind.
DISCLAIMER: I am not that familiar with your machine or the software/firmware setup.
If you can get to the ball screw on Y are you able to turn by hand, with machine power off?
Does the control allow you look at the real time logic state of different sensors i.e. limit switches, thermocouples etc.. A lot of times this will quickly allow you to narrow the scope of your search.
Is it an overheat I the true sense, which it sounds like it ain't. At least on the boat anchor I have deal with it is some times difficult to determine if the controller saying the servo pulling too much current during a given period of time and alarms out to prevent an overheat. Look at your schematic and test the component(s) that sense heat at the component level. If is okay work your way back, eliminating a wire problem. Ohm out the windings on the servo if they are all close in value move on. Meg the cable from the servo to driver.
Again not being familiar with your bit of kit makes it hard for me to trouble shoot from my sofa, but I had similar situation.
Very large HMC was telling me an axis was pulling way too much current. Servo ohmed ok, cable ohmed out okay as well. Day later after plenty of head scratching and consulting with greater minds that mine. I decided to repeat the meg on the servo to driver cable, low and behold the cable was leaking big time. So what was different between test 1 and 2 ? Rain yup it had rained a lot the night before test 2. Turns out our shop was built on land fill, the pit in which the machine sat just a titch above the water table. So when the 10 year+ cable which was run below the pit was now wet. Yha had to be their but what I am getting at is work from simple to complex in testing stuff, if someone else did a test repeat it and by all means use your test gear as an aide (Have meters, meggers etc. that are not crappy cheap 2 reasons: it won't blow up in your hand if you mess up and will give good data.). A lot of times part or all of problem is located between the driver's seat and the steering wheel.