I have a weird problem with my 9 year old Mac Mini. I came fairly close to just buying a new one, but found a solution. The machine frequently refuses to wake up after extended periods of sleep. Mouse movements and keyboard strikes fail to wake it. However, if I tap the power button it will wake. I then have to wait for it to go back to sleep before the mouse and keyboard work. It probably needs a new battery, but I am not putting any money into a 9 year old machine..
Sounds like a bad connection at the power button.