No, they weren’t sterile. The amount of known Neandertal DNA in living humans is roughly the same about they get from a great-great-great-great-grandparent.
There's a paleobiologist who is arguing that for modern Europeans to have as much Neanderthal DNA as they currently have the population must have started out as something like 50% Neanderthal.
That would make us the descendants of Neanderthals, meaning they never went extinct but just got bred into a larger population where many of their traits were selected against over time.
I wrote “most them were sterile” - as far as my understanding, this was the case. If you have (as most likely) more information, I’d be glad for links