Cat tested Positive & Negative Felv test

My sister had her cat tested for Felv leukemia today. The first test came out positive and the vet did another test and the second one came out negative. Should my sister be concerned? Which one should she believe.... the positive test or the negative one? 

False positives are more common than false negatives, because of the way that test works. So my guess would be that the negative is the correct one. But there is a more sensitive test, that needs to be sent to a lab, that she could do if she's worried. What did the vet say?

The vet said that the cat has a clean bill of health. And she showed my sister the tests showing two red lines or something that says its negative and the other with one red line showing positive. What is that more sensitive test you are speaking of? what is the test called?