One common problem is "integral windup". It might take too long for the output value to ramp up to the necessary value when the loop first starts up. Sometimes this can be fixed with a more aggressive differential term. Sometimes the loop has to be "preloaded" with a starting output. Another option is to disable the integral function until the measured variable has entered the proportional band.
Some PID loops control a valve or similar mechanical device. Wear of the valve or device can be a major maintenance cost. In these cases, the PID loop may have a "deadband" to reduce the frequency of activation of the mechanical device. This is accomplished by designing the controller to hold its output steady if the change would be small (within the defined deadband range). The calculated output must leave the deadband before the actual output will change. Then, a new deadband will be established around the new output value.
Another problem with the differential term is that small amounts of noise can cause large amounts of change in the output. Sometimes it is helpful to filter the measurements, with a running average, or a low-pass filter. However, low-pass filtering and derivative control cancel each other out, so reducing noise by instrumentation means is a much better choice. Alternatively, the differential band can be turned off in most systems with little loss of control. This is equivalent to using the PID controller as a PI controller.