Can people with poor mental health not just benefit from meditation, but realize its goals?
Some people with a mental health history have actually benefited from meditation.
Such benefit is known to occur on a case-by-case basis, similar to how not every person that lives as a monk benefits from monkhood.
Many meditation centres have experience with mental health and generally require approval from a persons's psychologist or psychiatrist before permitting attendance on meditation retreats.
I have read many sincere accounts of people reducing mental health issues with meditation.
Also, the atmosphere of genuine metta (loving-kindness) at Dhamma centres is also helpful in reducing anxiety & other self-esteem issues.
The loving-kindness of a Buddha radiates upon all beings, neglecting none (including fake monks).
While this answer focuses on mindfulness-based medical therapies that are derived from Buddhist meditation techniques, it does not directly promote original Buddhist meditation practices, for medical therapy.
This Feb 2015 article from Huffington Post reports that:
Mindfulness research pioneer and founder of Mindfulness-Based Stress Reduction, Jon Kabat-Zinn, who was in the audience at Davos, stood up at the end of the conversation to share his thoughts on the mindful revolution in mental healthcare, which he noted has been well underway for several decades. As Kabat-Zinn explained, research and testimonials from patients and clinicians suggest that we can turn "the medication down and the meditation up."
"We’ve seen this in the clinical domain for many years. People, in concert with their physicians... actually going off their medications for pain, for anxiety, for depression, as they begin to learn the self-regulatory elements of mindfulness," said Kabat-Zinn. "They discover that the things that used to be symptomatically problematic for them are no longer arising at the same level."
Patients, in concert with their physicians, could seek to try two therapies derived from Buddhist mindfulness meditation techniques, namely, the Mindfulness Based Stress Reduction (MBSR) and Mindfulness Based Cognitive Therapy (MBCT). More information can be found here and here.
Another interesting book by Kabat-Zinn is "The Mind's Own Physician: A Scientific Dialogue with the Dalai Lama on the Healing Power of Meditation".
Mindfulness-Based Cognitive Therapy (MBCT) is designed to help people who suffer repeated bouts of depression and chronic unhappiness. It combines the ideas of cognitive therapy with meditative practices and attitudes based on the cultivation of mindfulness. The heart of this work lies in becoming acquainted with the modes of mind that often characterize mood disorders while simultaneously learning to develop a new relationship to them. MBCT was developed by Zindel Segal, Mark Williams and John Teasdale, based on Jon Kabat-Zinn’s Mindfulness-Based Stress Reduction program.
The three founders of MBCT and Kabat-Zinn also published a book called "The Mindful Way Through Depression: Freeing Yourself from Chronic Unhappiness".
The recently held 10th Global Conference on Buddhism (17 - 18 June 2017 in Toronto, Canada) had the theme of "Neuroscience and Mental Health: Making a Mindful Connection". You can find videos from it on their Youtube channel. There were three neuroscientists who presented on Day 1, Session 3. You can find a video of that here. There were speakers from both the sangha and the academia - you can find a list here.
"Can people with poor mental health not just benefit from meditation, but realize its goals?"
Yes, if their poor mental health is not an obstacle to calming the mind.
Otherwise, the more the poor mental health is making the mind wonder, the harder to realize the goals of meditation.
The question is whether people with mental problems can benefit from the practice of Dhamma? Yes, in all cases.
Note that everyone has mental defects till reaching Arahat-hood.
It is important to take the medicine rightly and completely, since meditation alone also harms those "without" defect.
Here is a good talk about why it seems for some that the practice of Dhamma does not work:
The Healing Power of the Precepts, by Thanissaro Bhikkhu (1997; 4pp./11KB)
Many people today have come to the Buddha's teachings in search of emotional and spiritual healing. In this short essay the author reminds us that the single most effective tool for healing a wounded heart may be found.
It might be that certain mental "real" disabilities (vitalingo -- persons with congenital defects such as idiocy, one of the few conditions that make the goal impossible for one in this life -- see here about conditions) are hindrances to reach any path or fruit in this life time; but even then, in the long term it will help anybody who takes the medicine right and associates with people of secured faith in and of the path.
So (at the risk of contradicting other answers) I will always encourage people, with whatever kind of mental defect or defective mental state: to turn to Dhamma or try to get the best possible integration; to serve and to make merits around the Gems, and to practice Dhamma as well as possible; rather than to turn to usual ways, which usually just try to make one function in and for society, or to keep the "defect" prevented/protected from functioning and vice versa.
(Note: this answer is not meant for any commercial purpose or other wordily gains.)
There are other answers to this question, saying "Yes, in all cases" and "benefit is known to occur on a case-by-case basis" -- I'd like to agree with both.
There's a wide range of "poor mental health" and "mental illness".
In some of the more extreme conditions you might not understand people, see them as a devil, not be able to pay attention because of the voices in your head, and so on. I'm told that sometimes people need psychiatric medication first, before they can make sense of conversation ... but that (medication) isn't enough, what comes after that? The following are several ways in which I think it could help.
People with mental health problems may smoke, drink alcohol, abuse drugs, at a higher rate than the general population. A more Buddhist environment will tend to do without that.
People in general society may be judgmental, hostile; intolerant of people who are different; have expectations. Some characteristics of an ideal Buddhist might make them more tolerant, easier to be with, including: self control; good will; not necessarily wanting to profit from another person; not having too-rigid attachments or views to how other people "should" be; not seeing themselves as superior.
People with mental health problems may behave differently and not as you expect or want them to. If you're a normal, successful person you might live in a social environment where people do what you want them to do when you tell them to; and you may not know how to cope with someone whose response is different or indifferent. I think that "they drive me nuts" or "they make me crazy" or "they make me angry" is a not uncommon reaction people have -- and they might feel that their anger is justified (justified by the other person's unexpected misbehaviour). You may need to be a bit unusual, or have an unusual attitude, to practice unconditional good behaviour. Buddhists aren't the only people who might try to make their good behaviour unconditional, but it's explicitly a mental state that Buddhists train themselves towards and value.
In summary I suppose I'm saying that where you find the dharma you may also find the sangha ... and that the sangha might help, or at least be less hurtful than some other communities or societies.
And, yes. Some people (perhaps including monks) see society itself (or people's reactions to society) as a bit crazy. So far as I know, in general, a "sheltered" environment helps everyone.
I'd like to emphasize that it can help: be part of the solution, or at least not add to the problem. However, also: "You yourselves must strive; the Buddhas only point the way."