Some channelers seem to survive the taint for years while being able to function to a degree like Rand & Logain however I think there were reports of men who were at the Black Tower who went mad after their first touch of tainted saidan. Even if most don't go mad straight away from Taim's report as far back as Path of Daggers quite a few Asha'man had to be killed because they went mad and the Black Tower had not been going for a very long time at that point so the men who were killed could not have been channeling very long either.