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In Chapter Thirty-Eight of Order of the Phoenix we have the following passage:

Snape stared at him.

“Put that wand away at once,” he said curtly. “Ten points from Gryff —”

Snape looked toward the giant hourglasses on the walls and gave a sneering smile.

“Ah. I see there are no longer any points left in the Gryffindor hourglass to take away. In that case, Potter, we will simply have to —”

This seems to indicate that a house cannot have negative points. Once it hits zero no further points can be deducted. However, in Chapter Eight of Half-Blood Prince it seems that it is possible to have negative points:

“Fifty points from Gryffindor for lateness, I think," said Snape. “And, let me see, another twenty for your Muggle attire. You know, I don’t believe any House has ever been in negative figures this early in the term: We haven't even started pudding. You might have set a record, Potter."

So which is it?

Can a house have negative points, or can't it?

2

Probably not

I agree that there's a discrepancy in what Snape is saying in those quotes. However, I think it's much more likely that negative points are impossible. Otherwise, his comments in Order of the Phoenix make no sense.

“Ah. I see there are no longer any points left in the Gryffindor hourglass to take away. In that case, Potter, we will simply have to-”

If having a total of no points isn't grounds for not being able to take away points then why bring it up? (I know Snape is being spiteful but vocalising the fact still seems redundant). And why would he say "in, that case Potter, we will simply have to...?" if all he was planning on doing was taking away the points as he originally intended? People don't usually say "in that case" when they are planning on acting as they always intended.

Furthermore, I think it's highly unlikely that the Gryffindor total would be exactly 0 in the circumstances at the end of Order of the Phoenix. Not only had Harry and the other members of the DA been losing Gryffindor points from teachers at a previously unheard of rate but Gryffindor had also been targeted by the Inquisitorial Squad (a group of students consisting solely of Gryffindor-hating Slytherins). We know that there was a lot of petty points-taking going on that year.

“It’s only teachers that can dock points from Houses, Malfoy,” said Ernie at once.
“Yeah, we’re prefects too, remember?” snarled Ron.
“I know prefects can’t dock points, Weasel King,” sneered Malfoy; Crabbe and Goyle sniggered. “But members of the Inquisitorial Squad—”
“The what?” said Hermione sharply.
“The Inquisitorial Squad, Granger,” said Malfoy, pointing toward a tiny silver I upon his robes just beneath his prefect’s badge. “A select group of students who are supportive of the Ministry of Magic, handpicked by Professor Umbridge. Anyway, members of the Inquisitorial Squad do have the power to dock points...So, Granger, I’ll have five from you for being rude about our new headmistress...Macmillan, five for contradicting me...Five because I don’t like you, Potter...Weasley, your shirt’s untucked, so I’ll have another five for that...Oh yeah, I forgot, you’re a Mudblood, Granger, so ten for that...”
(Order of the Phoenix, Chapter 28, Snape's Worst Memory).

Given how many points had been lost that year I would find it highly surprising if the Gryffindor total was exactly 0. It'd be more likely to be a low positive number or to already be a negative number. Basically, if it were possible to take more points from Gryffindor then someone else would've done it already. It would not be impossible that, after all the additions and deductions of the year, there would be exactly 0 points (which is what Snape describes). It's just very unlikely.


Conversely, in the other scene, it isn't clear that Snape actually takes away the points. He says he does, of course. But I would suggest that Snape cares much more about gloating at Harry, annoying him and making him feel as miserable as possible than he does about the points. He knows that Harry values the points and can't tolerate injustice. Taking away the points was a way of getting to Harry, as Harry himself knew.

Still Harry remained silent, though he thought his chest might explode. He knew that Snape had come to fetch him for this, for the few minutes when he could needle and torment Harry without anyone else listening.
(Half-Blood Prince, Chapter 8, Snape Victorious).

I think it's entirely plausible that Snape knows that starting the term on negative points is impossible but counted on Harry not knowing this and so used the situation to take away points - simply to humiliate and infuriate Harry. He did it because he could, and it wouldn't exactly be the first time he'd abused his position in order to score a petty victory over Harry.

Even if this isn't the case then Snape's statement can be explained by him simply forgetting that negative points are impossible.

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    On the other hand, there were faculty members who were awarding points to Gryffindor on the slightest pretext, in order to undo as much of these shenanigans as they could. – EvilSnack Jun 6 at 4:34
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A House can Have Negative Points

To support this side, I turned to the Harry Potter Fandom Wikia page on House Points:

In the Harry Potter and the Half-Blood Prince, Severus Snape takes 70 points from Gryffindor before term starts, saying that Gryffindor would have negative 70 points, but in Harry Potter and the Order of the Phoenix, Snape was unable to take points from Gryffindor when they had no points left. It is possible, however, that Snape's next words after "In that case, Potter, we will simply have to..." would have been to send to Gryffindor into negative points, but he was unable to get them out before being interrupted by Professor McGonagall. It can also be assumed he was being sarcastic or joking cruelly about the situation.

So, really, perhaps Snape was going to put Gryffindor into the negatives in this situation, but McGonagall stopped him from doing so!

A House cannot Have Negative Points

This part is pure speculation based on events from the book. We see in Harry Potter and the Sorcerer's Stone:

At first, Gryffindors passing the giant hourglasses that recorded the house points the next day thought there’d been a mistake.

Apparently, House Points are recorded with an hourglass, and displaying negative points with this doesn't seem entirely possible!


So, I'd say negative points being possible is more favored through canon and other Harry Potter resources...

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