My two MiM J-basses (20-ish year-old 5 string and new 4 string fretless) have the slight thickening going toward the headstock just ahead of the nut. I never really thought about it much until I read this post and I had to go check. I guess I don't position my thumb that far down for it to effect my playing.
My Warmoth P-bass neck has a similar thickening, but it's just behind the nut instead of just ahead.