We can't really trace Indo-European back further than southern Russia of about 5,500 or 6,000 years ago, while current best evidence is that the last Neanderthals lived around Gibraltar or Portugal about 25,000 years ago, so it would appear to be quite a stretch.  People can sometimes retain purely oral folk-memories of things that happened thousands of years ago, but such memories tend to be heavily-genericized and assimilated to common motifs or archetypes.  Considering that Germanic folk-memory transformed Attila the Hun from an oppressive and sinister foreign conqueror to a noble king (Atli/Etzel) in less than a thousand years, I doubt that there's much possibility for accurate or specific folk-memories of Neanderthals over more than 20,000 years...