Actually, all medical tests can give false positives and false negatives. However, generally the consequences of a false negative are more severe than that of a false positive, so the cutoffs for what counts as a "postive" or a "negative" are set so that false negatives are much rarer than false postives.
I don't know about the tests for FIV or FeLV but I know most human medical tests give false postives more often than is generally assumed. Addiditonally, what most people (inlcuding, unfortunately, many doctors) don't realize is that the odds of having a disease (given a false positive) depend very heavily on your prior risk for a disease. To take a human example, a promiscuous gay man who tests postive for HIV has a much higher probability of actually HAVING HIV then, say, a man who's been in a mutually faithful relationship for 20 years because the first man's prior risk is higher. In cat terms, that means a cat who's been heavily exposed to FEV and FeLV (a recent feral, for example) is more likely to actually be ill (given a postive test) than a cat who has been an indoor-only housepet all its life.
That was probably more than you wanted to know. The upshot is that generally any time you get a postive medical test for something -- particularly something that it seems like you (or your cat) shouldn't really be at high risk for -- you should get retested before doing anything rash.