It's related, but it's closer to "stay in shape" than "stay healthy," which would include (in my opinion) a good diet and healthy life choices, as well as some good luck.
For example, a teacher who works with little children (a lot of coughing and not so much hand washing) might have a hard time staying healthy throughout the winter (catching a cold transmitted by her students), but has no trouble time staying in shape/keeping fit with daily visits to the gym.