This sounds like a common problem that we run into many times during the heating season. It is especially common for people with heat pump systems. A heat pump system typically has two different heat sources, the primary heat and the secondary heat. The primary heat is supplied by the heat pump which is just an air conditioner in reverse gear. The secondary heat is usually electric heat strips similar to the plug in space heaters you find in many retail stores. This type of secondary heat is what is called radiant heat and is very inefficient. You can tell how frequently your system uses radiant heat by how high your electric utility bill gets in the winter.
This secondary heat is sometimes referred to as emergency heat or AUX heat on your thermostat. It usually will have an indicator light or some other indication on touch screen thermostats. It can also be the primary source of heat if the thermostat is manually set to AUX HEAT or EM HEAT. Setting this mode will turn off the heat pump and only use the secondary heat. But this is very expensive so only use this if your heat pump is not working and until the service technician can get it back up and working properly.
In our case, what is happening is that the heat pump is not working and the system is relying only on the secondary heat. With a properly matched thermostat, the internal program recognizes when the heat pump is struggling to heat the house in cold temperatures and initiates the secondary heat to help the heat pump. The program also limits this help to when the actual house temperature is 2 or more degrees below the set temperature. This restriction, which is built into the thermostat program, is why the house will never get all the way up to the set temperature with the malfunctioning heat pump and the thermostat on the normal settings.
The situation does require further investigation as to why the heat pump is not working, but the information provided here is not enough to make a proper diagnosis. That will require someone to put their test equipment on the unit. The home owner can switch the thermostat to AUX HEAT in the meantime and it will keep the house at the set temperature, but it will be very expensive to leave it that way for too long.