Yes, when we look at measuring temperature with a thermocouple we look at the measurement plane. We have the thermocouple which has an uncertainty and we have the system reading the thermocouple which also has an uncertainty. The allowable error is the algebraic sum of the uncertainty. I can also order a thermocouple (or make my own) and have it calibrated and charted. Since a thermocouple can't be adjusted to some tolerance we chart them for calibration. I may use AWG 20 wire and request calibration at 5 cardinal points between 0 and 100 degrees C for example. I can then offset my display or indicating device for the known TC error at any point. I could for example also note my error at 150 C (using your temperature) and figure my error into my reading.
Thermocouples, while very good are not quite high accuracy devices. They aren't bad but aren't great either. If you multiples several each can have its own error. If I want to improve things I call a TC manufacturer and specify what I want and have my thermocouples made from the same spool of wire. Here in Cleveland, Ohio I dealt with Marlin Manufacturing prior to retiring. I called and they made me whatever I wanted. I also used Omega but Marlin was local and did good custom work. Anyway, yes, you change the thermocouple in a system and everything can change.
Depending on application other sensors can be used like the PRT as an example. A good PRT can be very, very accurate.
Ron