Kinda....
My understanding is that they wanted to put all handguns under the NFA, and therefore had to ban SBRs and SBSs because they were basically pistols. But then pistols got removed from the NFA, and nobody bothered to remove the SBR and SBR language.
That's why it's legal to have a handgun, but illegal to put a forward vertical grip on it.
The case (I think) you're thinking of is Miller, where he had a SBS, and the state argued that because the military didn't use them (they did) they were not protected under the 2nd amendment. But then Miller died and the case was decided by SCOTUS without him or any defense.
I probably have some details wrong, but that's the gist of it.