No, Sam Smith doesn't have Down Syndrome - but he did suffer from tonsillitis when he recorded the song "I Told You Know".
It's something he claims to regret, as the recording didn't turn out as good as he'd liked. But I think his voice sounds great on that track anyway!
The whole Down Syndrome claims are just people who are jealous of his success trying to be mean by spreading false rumours on the internet. Unfortunately these low-lifes exist, and it's best to just ignore them, and keep enjoying Sam Smith's talents instead.