Humidity is a funny thing.
The warmer that air gets, the more water it can hold as vapour. When that air cools down, it can no longer hold as much water vapour; if its humidity was high, then some of that water will condense out onto the coldest surface first. So air that has a healthy, comfortable relative humidity of 50%, when cooled, will have a relative humidity that increases as it cools, without any additional water vapour going into that air.
A room's temperature is also a funny thing
It's common in a lot of energy modelling of dwellings to assume that a room has a temperature. Studies looking at thermal comfort have to be a lot more sophisticated, because things like different radiant temperatures of different surfaces, and a big temperature difference between ankle-height and head height, can be uncomfortable. Most rooms don't have a single temperature - they have lots of different temperatures, clustered around the room's mean temperature. So if a room has a mean temperature of 16°C, there will be spots in that room where the temperature is warmer, and places where it's cooler.
Let's say you've got the bathroom at a nice comfy temperature when you're in it. And the kitchen's a nice comfy temperature when you're there. Maybe the bedroom, when you're not in it, gets down to 16°C. Now here's the thing - air moves around the dwelling. So of that lovely warm air in the bathroom or the kitchen, some of it will find its way through to the bedroom. Now, we put a fair bit of water vapour into the air just by breathing. Add in more water vapour from cooking and bathing, and the relative humidity in the kitchen or bathroom might get pretty high. And after all, relative humidity below 40% or so starts getting uncomfortably dry. Now, as that air gets to the bedroom, it's going to hit some surfaces that are colder than 16°C - 16 was just the average. Windows and window frames, external walls - particularly corners. It will often be the same spots, day after day after day - where there are thermal bridges: spots where heat has an easier route to the cold outdoors; spots that are a degree or three Celsius colder than other places.
Mould and Mites
Now, because those spots I mentioned above will be repeated sites of condensation, then mould will take hold, and flourish. It will keep coming back, again and again. And mould spores are not nice - exacerbating asthma and other respiratory diseases. Additionally, persistent high humidity allows dust mites to flourish, and their waste products also create and exacerbate breathing difficulties: they are "one of the best-documented environmental causes of asthma" (Peat et al 1998). Furthermore, people with Chronic Obstructive Pulmonary Disease (COPD) benefit from additional warming, and suffer in cold rooms (Osman et al 2008)
References and further reading
Oreszczyn T, Ridley I, Hong SH, & Wilkinson P (2006). Mould and winter indoor relative humidity in low income households in England. Indoor Built Environment, 15(2), 125–135. doi:10.1177/1420326X06063051
Preval N, Chapman R, Pierse N, & Howden-Chapman P (2010). Evaluating energy, health and carbon co-benefits from improved domestic space heating: a randomised community trial. Energy Policy, 38(8), 3965–3972. doi:10.1016/j.enpol.2010.03.020
Osman LM, Ayres JG, Garden C, Reglitz K, Lyon J, Douglas JG (2008). Home warmth and health status of COPD patients. The European Journal of Public Health Volume 18, Issue 4, 01 August 2008 p399-405. doi:10.1093/eurpub/ckn015
Peat JK, Dickerson J, Li J (1998). Effects of damp and mould in the home on respiratory health: a review of the literature. Allergy. 1998 Feb;53(2):120-8. doi:10.1111/j.1398-9995.1998.tb03859.x