They were around and contributing DNA up until more recently than formerly suggested.
Also, I've been watching some pod casts about this and the beginning date for Neanderthals’ presence in Europe gets kicked around a lot. It seems that their presence in Europe goes back further than initially stated and could conflict with the out of Africa story.
I'm waiting for some final consensus on that. I also saw something about a 500K year date for Peking Man that's also at odds with the out of Africa story.
There have been interesting speculations from time to time about relict populations of neanderthals lingering on much longer in isolated and mountainous regions, especially in northern latitudes. The speculations often point to the very ancient folktales found in so many places and across cultures about cave dwellers, trolls and giants.
It’s now thought that woolly mammoths survived on Wrangel Island until about 4,300 hundred years ago. That’s the Bronze Age. Achilles might have lived longer had he chosen to hunt woolly mammoths rather than Trojans. Unless the last surviving ogres and trolls got him first.
Anyhow, it’s not hard to speculate on a foundation for the ubiquitous myths and legends. Michael Crichton did a novel on it, and I imagine others have as well. A few movies have picked it up. To what vanishing tribe did Grendel and Grendel’s mother belong before Beowulf extinguished the last of the line?