It's an interesting question. The Toba Catastrophe Theory suggests that human population reached as low as 10,000 individuals during a climate change period linked to supervolcano eruption. Another theory suggests that human population reached as low as 2000 individuals. Overall I think 1000 individuals is enough genetic diversity that humans could recover reasonably well.
The real problem seems to me to be whether humans could ever catch up to where we are after being knocked down so low. Some people have suggested that if civilization collapses humanity won't be able to start a new industrial revolution due to depleted deposits of oil and surface minerals.
Oil (and coal, which is less topically sexy but historically more significant to industrialization) is the big problem, though rare earths and other materials that see use more in trace than in concentration could also be an issue. If you're a medieval-level smith, you probably wouldn't care too much whether you're getting your Fe from bog iron nodules or from the melted skeletons of god-towers in the ruins of Ellae-that-Was, although certain types of bottleneck event could make the latter problematic for a time.
Still, I'd be willing to bet at even odds that that wouldn't be a showstopper if it came to it.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.