I heard from a friend that used to work there that they were oddly averse to making mobile games- until recently all their games were Facebook-based. Draw Something is one of the few they actually prioritised as a mobile app, and it paid off.
This is the story! I was interested in this company early on because the products were great - I was playing the progenitor to this game, "Draw My Thing", four years ago - so I agree that there's more than luck going on here. For all their talk about "trying for years", it's telling that a search for OMGPOP on the App Store basically only yields three games.
Their main aim seemed to be to become a platform (albeit consisting of their own games) more than making hits. So they were reluctant to build iOS or Android-specific things (their early efforts were half-hearted compared to their impressive web games) and they were slow to build on Facebook as I recall.
