According to the "Appeal Process for 2009 - 2010 Testing Cycle", the "test scores and eligibility results will be mailed on or around January 29, 2010". But, according to that same page, if you want to re-test and appeal the eligibility decision, you must do that by "mid- to late- February."
This means parents have an extremely short window, only 2-3 weeks, to have a child re-tested and prepare their appeal. This window is so short that most places that can do that kind of testing (Woodcock-Johnson and WISC-IV) in Seattle may have no appointments available in the time parents have before the February appeal deadline.
The advice going around seems to be that, if you were not invited to the second round of district testing, then you can be sure you will get a negative result in the eligibility letter when it arrives. If you might want to appeal, do not wait for the letter to arrive, but book your appointment for private testing now.
Apparently, a similar situation happened in past years. This very tight APP appeal deadline seems inherently unfair if a mistake in testing has been made, likely to exclude otherwise eligible children, and particularly likely to exclude children of parents with limited resources and time. Please discuss it further here in this thread.
Update: A good discussion in the comments, not only on APP eligibility appeals, but also on other topics that might be helpful to people considering elementary APP next year.
Update: One parent, going through the APP eligibility process this year, writes:
We are still waiting on our letter! It just occurred to me that appeals are due 2/24, and as of Friday 2/5, we still don't have information. If we want to appeal and get a letter from our current teacher, we will have to get that from her ASAP because of mid-winter break coming up. This process is ridiculous!