Close to Awake

A philosophy professor told me he had a couple of Buddhist friends who were "very, very close to waking up." I asked him to visualize a scenario:

You're asleep, dreaming, when someone raises the notion of waking up. How interesting! So you attend a meeting of the Waker-Uppers, where the teachers, you're informed, are really close to awakening. These experts offer profound wisdom, and their placid, smug demeanors assure you that they're onto something super deep.

But just as you're captivated, a persistent buzzing abruptly intrudes. You open your eyes, slap your alarm clock, and trudge to the bathroom to pee.

So, those guys in the dream who were very close to waking up—what sort of timeframe are we talking? Weeks, months, or years?