Question:
When I move the "y" axis slide on my machining center close to the limit, the servo motor trips out. Is it likely to be the ballscrew or the servo?

Answer:
It is unlikely to be in the servo motor drive.
"y" axis slides on many machining centers are equipped with heavy sliding way covers that often get damaged by falling parts and tooling or just clogged with chips. When the slide moves toward it's limit the load on the servo motor exerted by the damaged cover can trip the motor.
Similar symptoms can be caused by an incorrectly installed ball screw.