Obsessive Compulsive Disorder: experiencing feelings of great anxiety when things aren't in a specific order (thats my own definition, but the actual is mostly just different wording). People with OCD can't cope with the world being out of order, so they stick to things they can keep in order- or they create in order to something that typically wouldn't have any order.
Dissociative Fugue/Amnesia: Mfkrs just wander off! Straight up, they don't remember who they are, where they're from, and they're often found 100s of miles from home. Why would someone do this? They can't cope with reality, so they escape it.
Anti-social personality disorder: You're anti-social because you can't cope with society the way it is. You can't deal (cope) with people, so you shut yourself off from them.
Anxiety: Anxiety is simply fear. Fear and anxiety are one in the same, physiologically speaking. Anxiety as a disorder is just fear that is disproportionate to the situation one is in. Essentially, its a result of one's inability to have normal levels of anxiety for the situation. If you're having grandiose fear resulting from a situation that seems "normal" to most, than you're having trouble coping with some aspect (if not all aspects) of that particular situation.
PTSD: You can't cope with a traumatic event which occurred in your past.
Body Dysmorphic Disorder: someone has a big nose (well, at least bigger than they perceive to be normal), can't cope with it- and all of a sudden they're taking shears to it trying to cut it off.
Anorexia/Bulimia Nervosa: Girl can't cope with the idea she could be seen as "fat" by her peers, she uses anorexia and/or bulimia to fix what she feels is a problem.
Suicide: Well, if you can't see an inability to cope with reality there... I don't know how else I can explain it, lol.
Though, not every Psychological "disorder" is a direct result from a failure to cope with reality as it is. Like I said, most of them are. When you get into schizophrenia, there are obvious etiological roots which lie in nature (genealogy) more than nurture.
Those are all just off the top of my head, but I could go on for pages if I actually dug up my old Psych notes. The way I see it, 95% of psychological disorders are simply failed coping mechanisms. However, there are also psychological "disorders" included in the DSM which are more physiological disorders dealing with chemical imbalances in the brain, its just easier for a mental health professional to diagnose/treat them than it is for an M.D.
Go ahead and ask any Psych professor you've met or will meet: how does inability to cope correlate with psychological disorders? And I'm sure you'll get a 5-10 minute lecture reiterating the same points I just made, only more articulated and wise.