Was Severus Snape a Potions student of Slughorn's?
By the timeline, it sounds plausible (Tom Riddle was, and he wasn't that much older than Snape).
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Yes, Snape was a student of Professor Slughorn's. As b_jonas states, Slughorn says so in Half-Blood Prince. Specifically, Slughorn says it here:
‘Snape!’ ejaculated Slughorn, who looked the most shaken, pale and sweating. ‘Snape! I taught him! I thought I knew him!’
Half-Blood Prince -- page 585 -- Bloomsbury -- chapter 29, The Phoenix Lament
One could also presume that Snape was Slughorn's student because Slughorn raves about Lily Evans's talent in potions and we all know Snape is Lily's classmate. There is no indication in canon that there were two Potions Masters teaching simultaneously at any given time in Hogwarts's history. That there is one teacher per subject at Hogwarts seems to be a theme (Defence Against the Dark Arts is the strongest indicator of that; one might argue that this is because the DADA position was cursed by Voldemort)
To add to this, Slughorn mentions he didnt think Snape could have brewed the Draught of Living Death better then Harry did, while Snape was a student.
“Stop skulking and come and join us, Severus!” hiccuped Slughorn happily. “I was just talking about Harry’s exceptional po- tion-making! Some credit must go to you, of course, you taught him for five years!” Trapped, with Slughorn’s arm around his shoulders, Snape looked down his hooked nose at Harry, his black eyes narrowed. “Funny, I never had the impression that I managed to teach Potter anything at all.” “Well, then, it’s natural ability!” shouted Slughorn. “You should have seen what he gave me, first lesson, Draught of Living Death — never had a student produce finer on a first attempt, I don’t think even you, Severus —”
This just adds to the fact that Slughorn taught Snape, and intimately knew his potions abilities in Snapes youth, a few chapters earlier.
Yes. Professor Slughorn explicitly claims so in HBP chapter 29.
Yes, we know that Lily was Slughorn's student since she was in his club, and she was Snape's age so yes, Snape was Slughorn's student.