If you want key fragments to drop for you, you are incentivized to not play Coops or Customs, champions from free rotation and, of course, you are stimulated to win your games.. but the matchmaking balance system makes you anyway win cca. 50% of your games regardless of your stimulation (unless you are superman or superlooser, in such cases the system might fail to balance your games :p).
Riot also stated as a bug if free champions don't grant fragments, but anyway, as long as it persists, you are motivated to not try out free rotation if you are after keys.
Lastly the amount of fragments per one key reset period is exactly 12, you cannot get more (Rito told us that chest and key limits were intentionally set to be equal).
Thus the only way how to open initial 4 chests (or more generally- any leftover chests that you have got at the moment of key reset) is either to buy keys or to benevolently not unlock some free chests from games.
Much easier access to chests and contrast between constant chest supply and variating key supply naturally leads to unbounded cumulation of chests over time for average player, but whatever..
It took me 161 mostly full premade games to get 12 keys last month, including:
18 Coops
0 Customs
1 Remake
71 matchmade losses
15 matchmade wins with (alas) free champs
56 matchmade wins with my champs
You can expect to win 50% of your games and cca. 55 qualified wins will yield 12 fragments-- you get your monthly asset in 110 legit games, hardly any sooner.
So if you are an average player, just play 4 legit games a day and you should be fine.
If you are an intensive player, you can play 15 games a day for a week, but then don't expect any fragment drops for the next three weeks.
Easier access to chests also effectively means that if you are an irregular player, you can expect your chests to pile up as well.