What people don't seem to know, is that your chain and cogs wear together. If your chain is worn out and keeps snapping, and you replace just the chain, then it'll have to wear into the already worn out cogs, and would barely be stronger than the old one. You're supposed to replace them all together, so that they all wear down evenly, which makes them last a lot longer! Seeing as you've gone from an old 17t cog, and now on a new(?) 16t, the chain would be worn down to match the old cog, so it won't run smoothly on the new one. It'd be best to replace the front cog, and the chain, but that's more money. But it's worth it! Or you could just have it skip for a while until it's all worn down...