So, I sent an email to api-support@battle.net last Tuesday regarding some timeout issues I was having on a development server (no response to that email yet), but it appears to have become much more widespread since then and is now affecting users of the TradeSkillMaster Application (http://tradeskillmaster.com/tsm_app). For some perspective, by our estimates, there are about 10k unique (and actively running) installs of this application.

We've had many people come into our IRC channel saying they are seeing issues with the application which we've determined to be related to one of the battle.net API's timing out. There are also a bunch of people reporting issues on our forums:
http://stormspire.net/official-tradeskillmaster-development-forum/13289-tsm-desktop-app- auctiondb-not-updating.htm
http://stormspire.net/official-tradeskillmaster-development-forum/13301-setup-wont-allow-me-pick-sever.html

Here's an example of a curl timeout I am getting from an authenticated request:
URL: http://us.battle.net/api/wow/auction/data/Aegwynn?
* About to connect() to us.battle.net port 80 (#0)
* Trying 12.129.242.40... * connected
* Connected to us.battle.net (12.129.242.40) port 80 (#0)
> GET /api/wow/auction/data/Aegwynn? HTTP/1.1
Accept: */*
Accept-Encoding: deflate, gzip
Host: us.battle.net
Date: Tue, 07 Jan 2014 8:51:52 UTC

Authorization: BNET <SNIP>

* Operation timed out after 10001 milliseconds with 0 bytes received
* Closing connection #0
http Code: 0
URL: http://us.battle.net/api/wow/auction/data/Aegwynn?
* About to connect() to us.battle.net port 80 (#0)
* Trying 12.129.242.40... * Timeout
* connect() timed out!
* Closing connection #0
http Code: 0

It seems that if I retry urls which are timing out a few times and then stop whatever program is making these requests, I will still be consistently unable to connect for some time afterwards (usually around 10 minutes). Even if I try connection to http://us.battle.net/en in a browser (or these forums, or any other us.battle.net website), my browser will be unable to and eventually time out.

EDIT 1/11 4pm PST:
http://dev.tradeskillmaster.com/timeouts.php

That page will display how many people have run into this issue that we've detected. It is as of r209 of the TSM application that we are tracking this issue, which was just released ~1 hour ago.