Yes, well kept fish can develop problems just like healthy people can get sick. However, the probability of experiencing any of the above with your fish is directly proportional to the quality of the environment your fish live in.
I can’t count the number of times I’ve seen cichlid keepers ask for help online with at least one of the above problems. It happens regularly. I can say with extreme confidence that most of the time the reason a fish develops a health problem is because of poor water quality. You hear experienced aquarists say it all of the time, “Do frequent water changes.” How much water to change and how often depends on many factors, and that’s a post for another day. But changing water weekly is a good rule of thumb.
In a healthy tank, the ammonia and nitrites should read zero ppm. ZERO. Sustained exposure to anything higher for either one will result in health problems and, eventually, death for your fish. Nitrates should remain below 40ppm, at minimum. Less is best.
Don’t be one of those who asks “How’d that happen?” Test your water regularly, don’t assume anything about your water parameters, and address any fluctuations immediately. You can’t prevent an occasional loss of fish, but you can minimize the frequency simply by being conscientious, doing regular water changes, and taking your fish keeping responsibilities seriously.