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Some people think that Veronica is a workaholic but what they don't know about her is that she actually suffers from a very debilitating and un-diagnosed chronic illness – she's really the opposite of a hypochondriac.

I am not satisfied with previous answers to this question.

Q: What do you call people who are misdiagnosed or judged by their doctors (especially in the US) as "hypochondriacs" when in fact they are suffering from real illnesses?

They are neither overly anxious or nonchalant about their actual condition. After years of being mis-treated, un-treated, mis-diagnosed, or un-diagnosed these sufferers go back into the workplace and stop talking to anyone about their condition because it is the most practical thing to do. They pretend they are perfectly healthy and silently suffer through their pains/symptoms. I don't know any hypochondriacs but I know dozens of these types of people.

Maybe the word I am looking for is the inability of doctors and society in general to cope with the actual sea of suffering/illness that there is in our society.

For the sufferers who are afflicted with chronic pain or hidden illnesses which may or may not be diagnosed by the medical profession, there is a term for this condition:

Invisible disability

Invisible disabilities are chronic illnesses and conditions that significantly impair normal activities of daily living. In the United States, 96% of people with chronic medical conditions show no outward signs of their illness, and 10% experience symptoms that are considered disabling.

And there's an interesting article online called Chronic pain: The “invisible” disability. The author describes the frustration of managing their chronic pain, long after the illness has been diagnosed and treated

…a study by the Institute of Medicine discovered that pain can endure long after the illness or injury that caused its initial onset has been treated or healed, until it eventually evolves, or devolves, into its own disease. That is, pain is no longer indicative of another prognosis — it is the prognosis, and a disabling one at that.

However, there doesn't appear to be a single word that describes the category of silent or invisible sufferers. The only single word that comes close is stoic, which Oxford Dictionaries define as

A person who can endure pain or hardship without showing their feelings or complaining.

If the OP described this group as the antithesis of hypochondriacs the meaning would be understood. Finally, the most common phrase appears to be living with chronic pain, which is universally understood and requires no explanation.

Anosognosia. Anosognosia, also called "lack of insight," is a symptom of severe mental illness experienced by some that impairs a person's ability to understand and perceive his or her illness. It is the single largest reason why people with schizophrenia or bipolar disorder refuse medications or do not seek treatment.