Curious, when did you start playing? Did you get your 1 chest per week for the entirety of last season?
You can only earn chests on champions you haven't yet earned them on this season. So last season, for the first 4-6 months people whined they didn't have enough keys to open their chests. I had 4 or so chests unopened for a good while. Eventually, I ran out of champions I could get an S on (and I don't queue with others, so I never get a chest due to an ally's S). I now have 12 excess keys due to the end of last season. I've since gotten chests on my 3 mains so far this season and haven't bothered chasing my 1 chest per week - I'll get them on the other dozen or so champions I'm capable of it on eventually, no point forcing it.
Anyways, I went a bit off topic. Point is, you'll probably run out of champions you're good enough to get S ranks on and in a few months eventually stop getting chests and just work up the keys. Also, you may be close to your monthly key reset which will make getting fragments a fair bit easier.