John Hinckley was a perfect example of someone who most definitely was insane and deserved to be acquitted by reason of insanity. Now in the aftermath of Hinckley's court proceedings, Congress and several states made it next to impossible to effectively use the insanity defense, which resulted in many of the prison mental health problems we have today. People don't understand that being found not guilty by reason of insanity is not the same as being found not guilty. In every state, and at the federal level, if you're found insane, you'll spend several years in a mental hospital instead of a prison. At least there you'll be treated for the illness that caused you to commit your crime.
Another good example is Andrea Yates. She was very clearly insane and therefore not criminally responsible for her crime of drowning her five children in the bathtub, but under the laws of the state of Texas, she was considered perfectly sane. If it weren't for a so-called "expert witness" for the prosecution basing his testimony off a TV show, Yates wouldn't have received a retrial at the federal level where she was rightly found insane and sent to a Texas state mental hospital.
Hinckley never was sentenced to jail. He was in a mental hospital, IIRC.