Very difficult to define what "mentally quite healthy" means. According to an article I read, one in every three people suffers some form of mental disorder (dissociative disorders, mood disorders, anxiety disorders, psychotic disorders, eating disorders, developmental disorders, personality disorders, ambulatory disorders etc etc as defined in the guideline criteria listed in the ICD, DSM and other manuals) at some stage in their life.
Empress Elisabeth did suffer from an eating disorder as well as probable anxiety disorders and depression and Crown Prince Rudolf appears to have suffered from depression. However, it was widely accepted that the official finding "Suicide by revolver in a state of temporary derangement" was, as aor states, a ploy to ensure a Catholic burial, as in those days the Catholic Church claimed that anyone who deliberately killed him/herself and was aware of what he/she was doing was a sinner who was not worthy of a catholic burial in consecrated ground. I am of the opinion that the Crown Prince knew what he was doing when he pulled the trigger...