One correspondent, Dr. W. Genck, was correct in his concern over the system suffering from a distance/resistance loss.
If the existing plant instrumentation was, as suggested by Dr. Genck, old, then it probably would not be of high impedance; which could certainly produce large errors. That does not represent, however, the substance of my concern.
Being retired I have no access to facilities whereby I could check the precision of my statements (it requires access to appropriate (NIST?) standards), so my memory might be suspect.
I believe that the required accuracy for commercial thermocouples is ±3.0°F and for premium couples it is ±1.5°F. In order to achieve any benefit from use of premium couples it would also be necessary to use premium extension (compensating) cable which is covered by the same standard. In any event, inspection of the cold junction compensation method/equipment is warranted. Last time I checked with a manufacturer (about 25 years ago) their auto-cold junction compensation accuracy was ±1.5% of full scale.
If the measurement is by RTD then similar problems exist: last time I checked commercial standard RTDs were required to be better than ±1.0°C and premium items ±0.5°C. Other problems can arise with using RTDs according to whether the measurement loops are 2, 3 or 4 wire (whereby the accuracies due to line lengths are respectively large, reduced but affected by Wheatstone Bridge arm ratios, and minimized.)
The major problem I see with either RTD or thermocouple primary devices (and compensating/extension cables) is that the above are stated (my memory limited) as "plus or minus" — thus you could get lucky by having both devices of same magnitude and polarity error, but that is highly unlikely (equivalent of 3¬4 standard deviations?). In fact, unless both high and low temperature devices (and compensating cable?) were check calibrated with high-accuracy test equipment (and skilled test technicians?) then, as I see it, the probability of the measured differential being anywhere +1.0°F is extremely low. They could be looking at a control loop thought to be controlled at +1.0°F and actually being up to 5.0°F. I do not understand the problem of nucleation of the process materials, but it seems like the existence of such errors could be the cause?