Sick turbines are just like sick piston engines (old car with plenty of wear does not accelerate as fast as a brand new one).
As turbines wear, they don't produce as much power for a given turbine temperature.
This is most noticeable in a 331 for instance when one engine is much newer than the other.
To match the torque, one engine might be at 640 EGT, while the other at 605. .
Once the turbine cannot produce the rated torque for a given temperature on takeoff it will be pulled off the wing. I never had much trouble with Garretts and TBOs however. They are generally pretty good once they are started.