The question is the title, sorta. It was asked by a classmate. Here is my answer.
The author of Hebrews believed that at a certain point, it was hopeless for the Christian who rejected Christ to be re-redeemed. This passage is not the only one – although there is a discussion on this passage in the original language as to which is the proper order in translation, and the proper sense of some of those verbs – to address this issue in this epistle.
We find that the premise of warning is the occasion of the letter. It starts with Hebrews 2.1-3. The picture here is clear – if the Law of Moses was strict about rejection, how much more do you think the Gospel is? One is not just rejecting the stone tablets now, but also the Divine Blood. This must not be thought of as backsliding either. Notice the passage here and the beauty of the words. Draw closer, drift away, the whole of verse 2, and neglect.
Notice 3.7-19 which is a warning against unbelief, disobedience, and hardening of the heart. The imagery is powerful. What happened in the wilderness? They suffered and lost their life. Here, the soul is at stake. Then in 4.1, the warning is made again about coming short of the Promised Rest. 4.11 notes that an effort is needed so as not to fall. The Rest of God is our goal, but if we will not work towards it, we will fall.
This passage in chapter 6 also mentions the word ἀδόκιμος (Heb 6:8). This word was applied to metal objects, such as coins, which were not able to hold the image which had been stamped upon it. These objects were worthless. In 6.8, the author says that those who reject Christ after knowing Him are worthless.
I make a brief note about the ‘how much more’ comparisons throughout the book. The OT was good, but how much more so what Christ has brought. The same is said of the punishments under the OT, but how much more dreadful for those who reject Christ.
I note then 10.26-39 wherein the author notes that deliberately sinning after Christ destroys us with Christ. Compare this notion of deliberate sin with Numbers 15.30, where the sin of the hand held to heaven requires that the person be cut off from the Israel.
Finally, I note 12.15-17 as well.
So yes, according to the Epistle of Hebrews, those who reject Christ – this is not backsliding – after having come to know Him will not enjoy eternal salvation.