A somewhat traditional or literary word for a bedroom or sleeping room. In modern everyday speech, 寝室 (しんしつ) is more common.
祖母は二階の寝間で休んでいる。
My grandmother is resting in the upstairs bedroom.
旅館の寝間は畳敷きだった。
The inn's sleeping room had tatami flooring.
Compound of 寝 (ne, 'sleep') and 間 (ma, 'room' or 'space'). The exact historical derivation is uncertain, but the spelling is conventionally associated with a sleeping room.