Well, this thread is in General. So I think it is helpful to consider how various sorts of approaches to when does a player roll and what is at stake when a player rolls can contribute to, or undermine, a player's sense of their character's competence.
The thread being in General also means that the contrast between "random event" and "function of a skill check* isn't self-evident. One way to establish "random events" is to have them as the consequences of failed skill checks. Even some versions of D&D use this method: see eg the example of a skill challenge in the 4e Rules Compendium, and especially the consequence of the final check - which is a failure - in that example.