First off, I don't think I see the phrase "mental illness" as most people do. My mind immediately says schizophrenia, schizoaffective disorder, psychotic disorders, Bipolar 1, severe depression, and the intense forms of PTSD/borderline personality disorder. If one says "How about anxiety disorders, other personality disorders, adjustment disorders, ADHD, etc" I sort of shrug and acknowledge that those count under the official definitions, and certainly cause pain and have a large impact. Having had episodes of mild depression (and one that was probably moderate), plus obsessive-compulsive disorder I am aware that even these cause suffering. Yet when I see research about "mental illness," part of me says "those aren't the same." This is likely because people soldier on through those somehow, lives impaired but not ruined. A percentage of people manage to carve out lives with the more serious disorders too, but the numbers are worse and it's very difficult. If you know any, congratulate them often.
I had to dig back through multiple news articles to get to this study reported in Science Direct about a correlation between intelligence and mental illness. Interestingly, the news articles all focused on the greater incidence of anxiety disorders (20% vs 10.9% in the general population) rather than on mood disorders, which had stronger numbers (26.8% 9.5%). Bethany will be pleased to see that the confidence intervals are displayed prominently.
There were also elevated rates of environmental allergies, food allergies, and asthma. The article gives a nice summary of some recent related research and a description of the hyper brain/hyper body theory they offer as an explanation. Also of interest: the incidence of professionally-diagnosed autism spectrum disorders was only slightly higher than the general population, but the self-diagnosis rate was high. FTR, I think relationships are going to show among autism, anxiety, OCD, and probably a few other things, but those aren't nailed down yet and I keep autism separate in most discussions.