Absolutely!
Maybe not, "Doesn't know", but isn't aware of.
Yeah well, doesn't know...isn't aware of...close enough. I think we can too.
Job 33:15-17
15 In a dream, in a vision of the night, when deep sleep falleth upon men, in slumberings upon the bed;
16 Then he openeth the ears of men, and sealeth their instruction,
17 That he may withdraw man from his purpose, and hide pride from man./
...and sealeth their instruction? What's that mean? Does He hold it back from our conscious mind? Maybe. We read during the day, and at night, sleep on it, and something happens. It get assimilated into our spirit and becomes knowledge. Then we know it, but may not realize that we know it. That's why it's best to meditate on only godly thoughts and precepts and not anything which is negative.
Speaking in tongues works along these lines I think. There is supposed to be an interpreter when tongues are spoken in church, however, what about tongues in private prayer? When this happens, it is the Holy Spirit praying for us. The Holy Spirit knows what to pray for us better than we know ourselves. But we don't even know what was just prayed for...or do we?! M<aybe our spirit does know the interp? This could be and we don't realize it.