Have you ran a BLAST search on your primers/probe sequence to determine if they hit any other organism? Failing that, have you ran the false positives on a gel, if they are the expected size and have no cross-reactivity from the BLAST then the samples themselves could be contaminated. 

These were the previous primers/probes that were used before I got here and according to paper work left to me, they ran a BLAST search and they should be good to go. We have been using the same primers/probes for over a year, with no issues (they were used a week ago with no issues). Unfortunately, we do not have the equipment to run gels.

Today I am rerunning these samples, along with new samples, and a new set of known controls to see what results I get. 

The most likely answer is still contamination.  Run out your samples on an agarose gel to see the product sizes.  Clean ALL of your workspace and equipment.  Throw away ALL of your reagents and get new ones (water, buffer, enzyme, etc.).

I have tossed out all the reagents and have new ones. Everything has been cleaned and I worked in the hood. I got the same results. The new samples I ran also all came out positive and while some of the known negatives stayed consistent; there were still a few samples that were either borderline or positive. I am completely perplexed by this. I wish I could run gels to verify the product.