If you're heating with electric for example you can sometimes buy electricity far cheaper at night (demand is low and traditional production can be easily spun-down). Thus buying electricity all night and keeping the house warm, whilst using more electricity, could be cheaper than paying for peak rate electric.
Physics alone could explain it in very rare circumstances - board construction that expands in warm air closing gaps, cool air opens gaps and causes more cooling. Heat exchange rate from in- to outside is then possibly greater at lower temperatures. Houses aren't simple to model.
The rest is possible, but I'd file using expensive electricity instead of cheap electricity under the thermostat not doing a very good job, or more precisely the person setting it up not doing a good job of setting it up to run more cheaply.
I'm sure there are exceptions to this somewhere, but it doesn't seem common. Thermostats don't typically have a way to command anything besides on and off.
However, some systems do lead to this because of flaws in the the implementation. My ground heat pump, for instance, has a backup electric heat system that it will automatically, and unavoidably, utilize if it hasn't reached the target temperature within a set, relatively short period of time. So in my very well insulated home I do indeed see significant cost increases if I do temperature setbacks, as the recovery period sees more expensive/less efficient electric heat kick in, versus just incrementally using the heat pump through the day. Some fuel-based systems go to a less efficient high-heat stage in the same sort of situation.