AFAIK the chances for a player to get a key fragment decreases as soon as he/she gets each key fragment. How and when does it reset?
According to Wingfield, a Riot enployee,
Just want to help clarify some things about key drop rates:
Key drop rates are NOT linear and only wins have a chance of dropping a key. So you will earn key fragments much faster in your earlier wins than you will in your later wins. Or said another way, each incremental key fragment drop is harder than the last (on average... there is variance). This resets on a monthly basis for each player individually. The goal is to create a nice ebb and flow.
Grade does NOT impact key drop chance.
Premades DO increase drop chance.
Chests earned has no impact on key drop chance.
There is variance and randomness (with limits). It's unlikely that what happened to your friend will happen to you.
Hope that helps clarify :)
The big issue with the key drops that the communitie, was that people had a very low chance of getting one after wins. However, after the recent patches (6.8-6.9) the chance of getting a key shard has increased and is now available when winning a aram as well. (still have to win)