What's particularly annoying is that this error occurs even when I've just fired up iTunes. How can there be a "timeout" when I just started the program?!
Fortunately, there's a reasonably easy fix, one involving, believe it or not, the system clock. Here's what to do (in Vista--the steps should be similar in XP):
1. In the System Tray, right-click the time and choose Adjust Date/Time.
2. Click the Internet Time tab, then the Change Settings button.
3. iTunes is balking because your system clock is out of sync. Therefore, you need to synchronize it with an Internet time server. Alas, time.windows.com almost always produces an error, so click the Server pull-down and choose one of the other options. (I've had good luck with time-b.nist.gov.)
4. Click Update now and wait a few seconds. You should see a message confirming the clock synchronization. If not, try again, or select a different server and then try.
5. Click OK to exit, then head to iTunes and try your purchase/code redemption again. This time, it should work!
So, think this is another case of Apple messing with Windows users? Somehow I don't think this problem exists on Macs. Hmmm.