I dislike this advice. Very few players actually have the time to acquire and maintain the skillset for 16 different champions (hell, most professionals aren't able to maintain that many), and actually trying to keep a champ pool that massive will hamstring you as a player.
For most players, I think it's best to keep no more than six or seven champions over no more than three roles, with as many flex picks as possible. It's actually reasonable for most people to be able to keep that many champions in polished form, and it's quite unlikely that you'll be forced to play one of your two off roles (especially if you learn how to play support).
For instance, suppose I wanted to play Mid, Top and ADC. I could learn how to play Tristana, Lucian, Kog'Maw, Jayce, Lulu and Ryze, and I would have three-six champs for each of my main roles, but with only six total champions to learn and practice.