寝る and 眠る: going to bed and being asleep
Compare 寝る for lying down or going to bed with 眠る for sleeping, separating bedtime from actual sleep.
The time one goes to bed
昨日は十一時に寝ました。 says that the speaker went to bed or went to sleep at eleven yesterday. 寝る, read ねる, is used for bedtime in daily schedules.
疲れたので、ソファーに寝て少し休みました。 says that someone lay down on the sofa and rested for a short while because they were tired. 寝る describes lying down here without establishing that they fell asleep.
Sleep and inability to sleep
赤ちゃんが静かに眠っています。 means that the baby is sleeping quietly. 眠る, read ねむる, focuses on sleep and has the te-form 眠って.
早く寝ましたが、なかなか眠れませんでした。 says that the speaker went to bed early but could not get to sleep. 眠れませんでした negates the potential form of 眠る.
Recognize overlapping uses
Both 昨夜はよく寝ました and 昨夜はよく眠りました can mean that someone slept well last night. Do not reduce 寝る to posture alone and 眠る to a completely separate category.
眠いので、もう寝ます。 says that the speaker is sleepy and will go to bed now. 眠い describes the feeling, while 寝ます gives the intended action; the speaker is not already asleep.
Is 早く寝ましたが、眠れませんでした contradictory?
Show the answer and why
No. Someone can go to bed early yet be unable to sleep. Going to bed and actually falling asleep are different.