Board Game Arena

Is there any situation in which you *can* play the caravansery, but choose to play something else instead?

I'd like to know what situations those are, and why. I don't think I've ever passed it up when I could play it.

yeah, always play it. I'm a play the percentages kinda player, so this is my favourite card.
Even if for some strange reason you think you don't need it, then passing it to your neighbour seems wrong.

Conversely, if I get it in stage 2 and don't have 2 wood, then I kick myself. hard.

There are many instances where you should take something else over Caravansery. The most obvious example is that if you are in science, you should usually take School over Caravansery (and against good opponents also Library and Laboratory should be taken over Caravansery in many cases). Also in lower player counts (3 and especially 4p, where resources are abundant) it is often worth taking red over Caravansery, as the Age 2 military is the most important part of winning, unless you are in green. This is especially so, if your wonder is not resource heavy (e.g., Ephesos, Alexandria). I noticed that autotaking Caravansery is a very common intermediate level mistake and I won a lot of 3p and 4p games by just taking reds, which a Caravansery taker passed me.

Additionally, if you will play more you will notice that in some (admittedly rare) cases even after taking Caravansery you might have to play 7 or 8 resource cards in total and in these cases it is often more efficient to play just 6 or 7 resource cards in total+red/blue instead of Caravansery.

At 6 or 7 players (IRL), yes.
[Unless we're playing with the Armada expansion, in which case never]

At low player count, taking a caravansery is both good for you, and an effective way of denying a good card to the other players.

At high player count, this second part is far less relevant as you're penalising only one other player (and there are 3 of them so you're not even sure to deny this card to someone who really needs it), so it happens reasonably frequently that taking some other card (usually science or military ones) is just better. It's still one of the best card you can hope for in your starting hand, but it's no longer a "always pick it if you can" IMO.

If I’m following a science strategy I wouldn’t take it over a school/library

If you get a lot of resources in Age 1, but no manufactured goods or marketplace, you probably should take Forum over Caravansery. Even more important if your neighbors don't have many manufactured goods.

I would only reject it if I am Hali B and inevitably going science.