Luck? We don’t need no stinking luck!! (I hope you get the movie reference) 
We can only know about Jesus if we are told about him, right? Either verbally or in a written form. ‘Verbally’ would include an angelic proclamation.
The closer either of those methods comes to the truth as revealed in the bible, the better we will know about Jesus. To the extent those methods stray from the truth of scripture, the faultier will be our knowledge. None of us has perfect knowledge, of course, but clearly some distortions of the truth are more harmful to our understanding of God and His loving will for the world. If an angel appears to someone and announces for instance, that Jesus was a liar, and we believe and act on it, the results could be disastrous. How do we know that He is not a liar? The bible tells us so.
But the opening post question is 'Can we KNOW (not know about) Jesus without the bible. I read it this way “Is it possible for a person to have a spiritual experience of Jesus Christ, a relational experience, without having heard about Him in one of the forms verbal or written”. That is, direct, unmediated revelation?
That is totally up to Him. I think it would be rare. A book that all in itself cast me out of Calvinism to these more hopeful shores, called “The Reformed Doctrine of Predestination” by Lorraine Boettner, answered the question of why billions in the world have not heard the name of Jesus preached in truth and power, and are therefore to (according to the Reformed) be damned - the answer is “because God wants it that way”. To show his awful Power, etc etc etc. What a crock.
Most obvious, at least to me, is that the sovereign Love of God, His wisdom and Providential care of the universe(s) and each of His creatures, is unceasingly turned toward each man woman child at every moment of every day. I’m not going to wander into the question of suffering right now. Noone is abandoned, no one is lost. each heart is known fully. God in His goodness and severity will work out the issue of bringing true knowledge of His Son to each one, even after death.