My experience buying playoff tickets (elsewhere, obviously) as a non-ST holder:
I was a college student in Chicago when the White Sox went to the playoffs in 2005. I was able to buy ALDS tickets online by signing onto Ticketmaster.com at the exact moment they started selling seats, and repeated the feat for Game 1 of the ALCS. It absolutely helped that I was on a university internet connection, which was much faster than what most consumers have at home. Game 2 of the ALCS, I got shut out and had to get tickets from a wealthy friend whose dad had season tickets.
World Series? Forget about it: Ticketmaster's website crashed within seconds. A week before Game 1, the cheapest ticket was selling on the secondary market for maybe $700. This was in the nosebleeds. I figured I would wait and that prices would go down. I was wrong, prices got higher and higher as the game came closer. I ended up paying four digits for a seat literally in the last row of the upper deck, obstructed view behind a pillar, but it was worth it and I would do it again. I later heard that by the 3rd inning or so, you could get tickets really cheap from scalpers outside the park. Like only $100.
From what I understand, a lot of MLB teams will hold "lotteries" for single ticket postseason sales. This means that you can't rely on your good timing and your fast internet connection to get you a seat: you have to win a lottery.