Now she has two problems: First, the appearance of extreme physical weakness, which is not a quality voters look for in a President. That's the small problem.
The big problem is that she's just spent a year teaching Americans that they can't trust what she says about things like this. "She got overheated" is the new "there was never any classified information in my email, and I turned over everything work-related." We know, from recent experience, that she and her people only tell as much of the truth as they are absolutely forced to, and give up more truth only when there is no choice whatsoever.
The 'overheated' story was so implausible, in sub-80 degree weather without humidity, that we've already seen the first modification: 'she has pneumonia.' OK. Even if that's true, pneumonia is not terribly likely in the summer without some additional illness suppressing the immune system. What illness is that? She's been coughing for a long time.
The only thing we can be sure of is that she's not going to level with us about whatever it is. That's true for this, as it was true in the classified information scandal, and as it will be true with regard to everything ever if she's elected President. It's who she is.
The Democratic Party, were it wise, would take this opportunity to force her to stand down for anyone else. They could only improve their electoral prospects by doing so, and would in the meantime do the nation a significant favor.
UPDATE: Two questions I've seen asked and not answered: if she has contagious pneumonia, how could her campaign justify sending the little girl out to hug her in the "all is well!" photo-op? If she has contagious pneumonia, why would they take her to Chelsea's home, where there are babies who could be at risk from such a move?
There are a range of possible answers to those questions, but none of them make her look good.