There's only 2 diagnotic tests, and if true they'd both be susceptible to false-positive from a "dead" virus. Molecular test (PCR) - a super reliable scientific method that reads genetic material of the virus - and antigen tests - this test looks for the proteins on the surface of the virus. Diagnotic tests are supposed to tell you if you have an active infection. That's because you HAVE to have a viral body on the swab. Theoretically, I suppose you could have dead viral bodies in the nasal cavities from some time before; however, these athletes have been getting tested regularly so you'd suspect they would have tested positive before now if this were the case. Also, folks test negative days after recovery I believe. That leads me to believe there's either a natural flushing of the sinuses, or an element to the test that requires a "live" virus. Also, to rule out false-positives don't they retest to get a second positive?
The third test is an antibody test. That's the blood test that tells if the patient has any antibodies for the disease in question. This was big early on until, I think, they realized that the antibodies are so close to those for the common cold that we can't really tell them apart, and the prevalence of the antibodies deminishes after a relatively short time. So, it's not as reliable. (Pro: it appears are T-cells maintain memory, so reinfection is low/not severe)
This is all my understanding from my readings. Conclusion: they're probably using PCR, if not then antigen, and I don't think that's a huge issue. They should retest when there's a positive anyway, and after 7-10 days without symptoms they should be back.