It's more than likely that this so-called mysterious "weakness syndrome" is really just as a plot device to drive the story/characters along. This particular aim to draw sympathies to the particular character and is more often than not, a popular way to have character die before their time, if that's what the story demands.
So why is this mysterious weakness trope so popular, you ask? Why not always using a real illness? Typically it's because this "weakness" is a "clean" and "cute" illness, when compared to actual chronic medical conditions. Even in its lethal form the symptoms are: you are weak, you're getting weaker, you die. (coughing up blood every now and then is optional)
In contrast, an actual conditions like stomach cancer on the other hand is not "cute" at all. As soon as you specify a real-life disease/conditions a lot of questions pop up, something that might shift the perspective away from the tragic fate of a character.
Additionally, there might be some actual conditions that inspired the "weakness syndrome" since Japanese did have a lot of sick people for a long time in certain areas. Having been a country that's been devastated twice by an atomic bomb, the effects of radiation exposure was not well researched or documented back then by scientists, so a lot of the long-term from direct and indirect radiation exposure side-effects like birth defects, leukemia, anemia, cancers, thyroid disease, liver and bone marrow degeneration, and overall severely compromised immune systems, might have been attributed to other things.