So I have been keeping track of my key frags closely the past few weeks. Last month was the first month for me I was able to obtain 12-13 keyfrags (possible I earned a keyfrag from the previous month just before the reset, or I got an extra one from a bug when hextech crafting had to be rebooted for unexpected maintenance last week). The cap per 30 days is supposed to be 12 keyfrags. I have ceased earning any keyfrags since I logged 13 despite a dozen plus wins. I counted the number of victories I had in my match history between my last keyfrag and the first one last month and it was about 60 VICTORIES. Half of those were duo queue ranked. Most of the other half was solo aram. I believe my keyfrag date reset is about the 10th of July (it is supposed to be every 30 actual days, but I suspect it could also be 24 or 28 days too)
Yes it is supposed to be possible to keep up in keys with your box acquisition. But if you fall behind and your keys reset there's no free way to catch back up unless you stop getting boxes at maximum rate too. Currently I figure it takes a little over 60 wins to get 12 key frags a month; with a 50% winrate that's 120 games, which is 4 games a day. Luckily last month I play more than usual and had a strong winrate.
The first 6 keys are easy. You can get them in about 12 wins or so. The first 3 come almost in 5 measily wins.