Yes, it is very possible to have a false negative result- especially if your vets office uses a snap test. There are soo many variables that must be done correctly- if they are not preformed properly- you can get an inaccurate reading.
When the blood is drawn, (if they use a snap test) they put 3 drops of blood into a vial, add 4 drops of solution to it and then shake the mixture 5 times. Then they place it in the end of the snap test, wait until the blood mixture reaches the dot window and then snap the test down. If any of these steps are not done exactly right- it can cause errors in the test. Also- if your cat was under 6 months they would have only in most cases tested for FeLV, and would have not preformed a combo FeLV/ FIV test.
One more thing- if the snap tests (which are used in most cases) were allowed to get too hot or too chilled- it can also impact the way the tests results turn out. So yes, it is very possible and there are many factors that can be involved in a false neg or false positive result. This is exactly why vets and vet techs alike must be very dilligent when they preform these tests. And sometimes, even though they may be dilligent and do everything properly- you just wind up with a bad test. In my experience- if money is not a factor- it is best to test 2x for a more accurate result.
I am soo sorry your kitty is being pts hon
My thoughts and prayers are with you