Status dictated what kind of furniture and comfort each person enjoyed.
Monks slept in an unheated, large dormitory. Each had a straw mattress on a wooden bed-frame, with a single sheet and one blanket; they slept fully dressed except for their belts and knife-sheaths (see The Rule of St Benedict, chapters xxii and lv). Among the Cistercians, sheets were of linsey-woolsey (linen and wool mix) and mattresses of blue ticking, the straw replaced only once a year. At Durham cathedral priory it was the custom for this blue mattress to be carried in procession to the grave of a monk when he died, where it served as a canopy.
In his detailed work De nominibus utensilium, Alexander Neckham describes the furnishing of an aristocratic bedchamber in about 1180:
In the bedchamber let a curtain go around the walls decently, or a scenic canopy, to avoid flies and spiders. . . . Near the bed let there be placed a chair to which a stool may be added and a bench nearby the bed. On the bed itself should be placed a feather mattress to which a bolster is attached. A quilted pad of striped cloth should cover this on which a cushion for the head can be placed. Then sheets of muslin, ordinary cotton or at least pure linen should be laid. Next a coverlet of green cloth or coarse wool, with a fur lining of badger, cat, beaver or sable . . . all this if there is lacking purple and down.
The contrast with the bed of a monk could not be more striking.
Imagine that the poorest levels of society (that is the vast majority of the population) did not even have a wooden bed-frame, but slept on a palliasse on the floor, or even a pile of loose straw, with as you say clothing piled on top for warmth. Furniture was an unreachable luxury for the average manorial field-worker. Furs were strictly for the wealthy.
Today we are not familiar with the concept of extreme poverty, class distinctions or domestic hardships and privations but for many English people at that time these were the reality. "Being a bit cold" was normal for both monks and manual workers - the "labours of the months" illustrations in manuscripts often show poor people toasting their feet in front of a fire during February, for want of any other means of warming themselves: