There are several conversation achievements (listen to every Blacksmith, Jeweler, Cain, Leah, Tyrael, Adria, town merchants, followers, etc.). Is it possible to miss any of these by not talking to them at a specific time? If so, which, and when?
While you might be able to miss a required conversation when playing strictly linearly, you don't have to play strictly linearly.
Diablo 3 is broken up into quests, and since you can start from any quest you've unlocked, you can jump back to whatever conversation you have missed if you go too far, too fast.
Once you've unlocked all the acts in a given difficulty, you can jump anywhere you want, at any time, so it becomes less an issue of "missing" a conversation than it is "figuring out which conversations I've yet to have". (And then jumping directly to the relevant quest)
I would presume yes, it is possible to miss this achievement ( but I have not confirmed it). You do not need to talk to everyone to complete an act. Therefore, once an act is completed - I would presume you lose your chance to have a conversation about specific current events (removing your chance at that achievement).