I hope so, but my son definitely seemed to have several nightmares, waking visibly frightened and crying before he was 9 months old. I don't know what kind of nightmares he'd have, as I don't think I ever saw him frightened when awake, and he only cried when he was hungry. (Such an easy baby to take care of, though I needed to manually check his diaper. If he was crying, he was definitely hungry. Apparently many babies cry if they've wet their diaper.) His nightmare were presumably hunger and/or abandonment.
My son also seems to share my abnormal reaction to pain... beyond a fairly tolerable threshold, I go numb. I know I've injured myself somewhat seriously if I feel quickly rising pain followed by numbness. When my tooth got knocked loose, it went numb. The three times I've sprained ankles (once in the woods, 3 days form the nearest hill with cell phone coverage), my ankles have gone numb. The time the cartilage in my nose tore from the bone, it also went numb. Ditto for the time I burned my finger while arc welding (though, I think that time I killed the dentrite back to my spine and it had to grow back out). My son takes a tough fall and other parents look at me like they expect my son is going to cry, and my son just gets up and keeps playing. So, I don't think his nightmare were about pain, either.
He just spent his entire life in a nice cozy temperature controlled pool with a constant IV of nutrients. Now he's been thrust into this dry liquidless, cold, world with intermittent nutrition. Kinda sucks if you think about it.