"Your expected wait time is 45 minutes..." And then after a brief moment, the same voice comes on and says "We've cut your time in half. It's just 20 minutes now!"

There's been some fun research done on the topic of progress bars and user perceptible wait times. (To the surprise of absolutely no-one who is paying attention, a) people suck at math and b) if one has accurate knowledge that a process will take 100 units of time, coloring in 1% of the bar per unit of time is provably suboptimal with respect to user happiness.)

Disneyland does something like this with wait times for attractions.

According to a 1989 Newsweek article: The waiting times posted by each attraction are generously overestimated, so that one comes away mysteriously grateful for having hung around 20 minutes for a 58-second twirl in the Alice in Wonderland teacups.

Source: http://dspace.mit.edu/bitstream/handle/1721.1/37703/21793670...

IOW, as an earlier manager of mine would say, "Set low expectations, and exceed them."

I am still ambivalent about that, though.

My dad always says the way to make people happy is to under-promise and over-deliver.

and that's one of amazon.com's official mottos.

Always a nice moment when that happens. Though I believe your talking more about "under promise - over deliver" rather than applying a sort of progress to a certain goal. Always use "under promise - over deliver" when possible, your customers will love you for life :) (exception: DMV)

Like loading screens that go faster near the end? I don't know that making the wait time look worse than it already is (40m vs 30m) will please customers.

I agree - for a once in a blue moon interaction, sure, not having to wait as long as you originally thought might be good.

For any kind of interaction which may be repeated then it just makes you look like a stupid business owner or software developer who can't properly estimate how long something will take!

I'm not sure that's true. One case that comes to mind is World of Warcraft server queues. In general, waiting in a queue to log into a server is something a player does either daily or never, and people really like to comment on their very exciting experience in the queue once they finally get in. In several years, I don't remember ever seeing a negative comment about how the queue didn't take as long as the initial estimate and in fact there were often positive comments when that happened. On the other hand, when the initial estimate was too low, people frequently complained about how bad Blizzard was at providing estimates.