Depends on the shelter. We have lots of long-term residents (6+ months) and low turnover, and so they are usually pretty well-vetted by the time they go home...but even so, Upper Respiratory Infections are very common in all shelter cats, and we recommend an immediate vet visit before introducing the new cat. It's just the smart thing to do. We test for FIV/FeLV, spay/neuter, deworm, treat for fleas and ear mites, and provide a full set of vaccinations for every cat, so best bet is to ask the shelter what they do and what the cat still needs. We also do additional non-emergency work (like dental care) on our long-term residents who need it, as time, money, and the vet's schedule will allow.
Even so, I don't consider any animal to be safe "out of the box," and would highly recommend a vet visit/checkup before introducing any cat to your resident cat.