And the 4 - 6 month window between infection and testing positive?
Can be counted against the window in which people are saying they haven't had sex. Do you really think someone going through all this trouble to give blood has any real likelihood of lying about their very recent sexual activity?
As for the test itself:
"The most common HIV tests use blood to detect HIV infection. Tests using saliva or urine are also available. Some tests take a few days for results, but rapid HIV tests can give results in about 20 minutes. All positive HIV tests must be followed up by another test to confirm the positive result. Results of this confirmatory test can take a few days to a few weeks."
http://hivtest.cdc.gov/faq.aspx#tests
The few weeks part later is to narrow down false positives, not false negatives. If you test negative in the first test you don't have AIDS.