From an excellent blog, Neuroskeptic, here is a survey of some data on mental illness incidence and suicide rates across countries.  The correlation is surprisingly low:

So what’s the story? Take a look –


In short, there’s no correlation. The Pearson correlation (unweighted) r = 0.102, which is extremely low. As you can see, both mental illness and suicide rates vary greatly around the world, but there’s no relationship. Japan has the second highest suicide rate, but one of the lowest rates of mental illnesses. The USA has the highest rate of mental illness, but a fairly low suicide rate. Brazil has the second highest level of mental illness but the second lowest occurrence of suicide.

Perhaps I am reacting too much to the examples in the excerpt, but one possible explanation comes from noting that there is a difference between incidence of mental illness and detection.  In countries where mental illness is readily diagnosed you will see more reports of it and also (assuming it does any good) less suicide.  The US must be the prime example.  And as I understand it, mental illness is highly stigmatized in Japan so the effect there is exactly what the data would suggest.  I don’t know about Brazil.