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I am sorry to hear that your kitties are having issues with nausea, vomiting and diarrhea at the moment. The short answer to your question is that "yes" they could absolutely have passed something back and forth between them that is making them ill at alternate times. Other possibilities include something in the environment that they are both exposed to with one having a delay in symptoms compared to the other one. If they go outside, it may be that they each picked up different bugs that they have exchanged or if one goes outside then he could have brought it in to the other cat.
It may also be something they are eating. They may have eaten the same thing that made them both ill but one experienced delay in symptoms for some reason or one was later to discover it, explaining the difference in timing.
I recommend keeping them separate if you can until they are both symptom free, eating and drinking normally with no diarrhea or vomiting. This will help to break the cycle if it is something that they have passed back and forth. Otherwise, you can have them both evaluated by your veterinarian and treated for contagious things such as internal parasites, bacterial infections, etc. Unfortunately viruses do not respond to traditional antibiotics (there is no evidence supporting the use of antiviral medications for GI disease in cats currently) and they will have to ride it out if it is viral.
Have I been able to address your concerns tonight or do you have others?
The expectation would be that if it were a virus, the cat that got better would not get it again so soon but not every illness induces a strong immune response and it is possible to contract the same illness if re-exposed. The other explanation is that it was something that one gave another but the first one did not get over it completely hence the symptoms came back. Alternatively, the first one got over the initial cause of diarrhea but now may be having diarrhea for another reason all together.
Since there are many causes of diarrhea the cause of the initial bout may not be the same as the second bout.
So while its possible it was something contagious that they are passing back and forth it is not necessarily so.
If I had to guess, its both cats being ill I would suspect either a contagious cause with incomplete recover in the first cat or an environmental exposure that is shared by both cats that remains in the environment.
Insects, cleaners, detergents, soaps, the same pathogen (bacteria or virus causing illness), plants, etc. Basically anything in the environment that they may ingest, lick or come into contact with.
As far as the cause, no this does not tell us anything.