The ideal is to design the control system so it has an infinite update rate and infinite feedback resolution like an analog system. The integrator gain term for a temperature PID looks like this:
One can see that as Ti becomes large the integrator gain gain can be very small. In fact it may round down to 0 or the gains may only change in increments like 1,2 or 3 when a 2.33333... is required. This is poor gain resolution. This is a problem with any controller that uses only 16 bit integer gains and keeps track of only 16 bits of output. To keep from losing resolution the gain may require 32 bits where the extra 16 bits provide extra resolution, not range. If your PID doesn't use 32 bit integer or floating point math it will suffer from quantizing or loss of resolution problem. The resulting integrator gain will not be accurate and could make the system impossible to tune. One should use 32 bit math or floating point controllers to avoid this problem. Resolution and dynamic range is the key.
Poor feed back resolution is also a problem. As the sample intervals get smaller the calculated rates of change for the derivative get bigger. The derivative gain is calculated by
where T is the sample time. As T becomes small the resulting gain can become quite large. If the feedback is coarse, noisy or suffers from sample jitter then the resulting rate in change of error or PV will cause the derivative term to be useless. If the feedback is course then the output will change more for each change in the output. Noise is also amplified by the higher gains that occur when sampling faster. The effects of sample jitter become worse as the jitter becomes larger with respect to the sample time.
Slower sample times effectively 'time average' the derivative gain but this also introduces phase delay.
When sampling at a very fast sample time one should use fine resolution feedback that is sampled at even intervals and the calculations should be done using 32 bit math or floating point.
