I hope someone can find a better answer, however until Winds of Winter is published
I believe the answer is simply that George RR Martin intentionally left the reason vague.
Possible explanations include: Physiological
The persistent cold found The Wall impaired his movement Someone drugged him Side effect of the burn he received while saving Mormont's life.
Jon's chapters mentioned the burn regularly Magical
A spell of Melisandre Interference by the Old God's Skinchanging into his direwolf
Psychological (John's shock at the events) However
unless someone can find more supporting evidence I believe it's pure speculation until we get more of the story.
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This is being way over-analysed.
Swords stick in scabbards when cold.
This is true even in the real world.
In fact, it is mentioned in another part of ASOIAF where, whomever it was (maybe Jaime).
loosened their sword in the scabbard
so it would not stick - in case they needed it.
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From the text: Men were screaming.
Jon reached for Longclaw, but his fingers had grown stiff and clumsy .
Somehow he could not seem to get the sword free of its scabbard.
-A Song of Ice and Fire: A Dance With Dragons, Chapter Sixty-One (Jon).
This is unfortunately all we have, since the explanation is not out yet.
Stiff and Clumsy
It is brought up many times in the books that Jon has to "flex his sword hand" to keep it active, and as brought up by Memnoch , his fingers must have gone stiff.
Somehow
And then GRRM throws in a little bit of a spanner in the works.
The phrase ' somehow ' alludes to something else holding Jon back.
Suspects/Explanations: Melisandre : could have cursed his hand, but it would defeat the purpose of her warning him of "daggers in the dark" Warging : into Ghost is a possibility; his mind was already half gone, so it couldn't control his body very well Poisoning / Intoxication : could have stiffened his reflexes, although Jon did not have anything to drink that night Someone : holds the sword in the scabbard.
Remember, there were multiple attackers Not just a scratch : Jon is injured, severely so.
Most people believe that the first couple of blows were really light/weak because Jon puts up a strong fight # ; but he could easily have been more injured than we think We will not know the true answer until Winds of Winter or beyond.
# I for one subscribe to the " Jon the Berserker " theory, which outlines that for some reason (Wolf Blood or Dragon blood) he has moments of near super-human strength.
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I think Jon froze up because they took him by surprise.
I don't remember finding anything unusual about his reaction (or lack thereof).
He was being betrayed, it was clearly an ambush, and coming from people he basically trusted.
I think the point is that, in some way, Jon's still a kid.
He can fight well, and he can lead well, but he hasn't yet dealt with something like what was happening to him at that moment.
I think most people would freak out a little if they were in his place.
No poison or Melisandre magic necessary.
In fact, I think if in that scene Jon just sorta went "hayia!!" and started expertly hacking away at the people who were his friends a second ago, it would have been bad writing on GRR Martin's part.
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If Jon carried Longclaw on his back, then it would be physically impossible for him to draw Longclaw from his back.
Else it could be the sword got stuck in the scabbard (maybe he didn't clean any blood off it?)
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He was taken by surprise at close range by men with weapons better suited for fighting at close range.
Even if he had drawn it it would not change much.
Maybe he'd take a few with him, but GOT fight's are pretty realistic as far as such things go.
He'd still be stabbed to death.
This isn't an action movie where one guy can take 20 with no problem.
Also, swords can stick in their scabbards from cold Anyone who has watched Gladiator has seen Maximus take out two soldiers thanks to this.
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In  A Dance with Dragons :    Jon dodged the first attack with only a minor wound.  The second attack succeeded only because Jon's fingers were like stones and he couldn't get Longclaw out and then the kid got him in the belly.     Was there a reason for this?
Why couldn't Jon Snow pull out Longclaw?
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<a-song-of-ice-and-fire><jon-snow>
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Most people believe that the first couple of blows were really light because Jon puts up a strong fight; but he could easily have been more injured than we think. Other factors could be that swords can get stuck in scabbards when cold, or the element of surprise. There is as possibility John was shocked at the events which prevent him from pulling out Longclaw. Also, his movement might have been impaired, he might have been drugged or the burn he suffered might have impaired him. Perhaps Melisandre even cast a spell on him. Furthermore, it is possible the Old Gods interfered and impaired his ability or that he skinchanged into his direwolf.
John was shocked at the events which prevent him from pulling out Longclaw. His movement might have been impaired. He might have been drugged. The burn he suffered might have impaired him. Perhaps Melisandre cast a spell on him. It is possible the Old Gods interfered and impaired his ability. It is possible he skinchanged into his direwolf.