This is entirely wrong. You can suffer from depression (and anxiety) and attribute it to other things, such as personality traits. Many people think "that's just me" with those that know them thinking the same. For less acute forms of depression such as dysthymia, that's common. People can think they're entirely normal and they're just not good at certain things.
If you talk to sufferers of serious and disruptive mental illnesses such as schizophrenia and bipolar with psychotic features, most of them will deny they're ill at all and rationalise their behavior. They'll generally talk about it by saying "well my doctors say..." and suggest that their doctors have been mislead or have misdiagnosed them.
It's much more complicated than it seems. It's all part of the perniciousness of mental illness.