Mine is set on 67 and its keeping it 69 right now. I don't care how high the power bill gets, we gonna keep it below 70 if we have to put a commercial unit on the roof and a window unit in every other window.
I'd look at your insulation situation first. That can make a big difference. Even if all you can do is add it to the attic, it will help, perhaps a lot. Also look at your window and door seals.
When we lived in an 1909 house that leaked air like a sieve, we did have some trouble keeping it at the set point on the hottest days. I kept a note by the thermostat that said "Do not touch. A thermostat is an on/off switch controlled by a thermometer. Turning it lower will not make it cool down any faster." Otherwise, sometimes I'd wake up in the morning and the house would be 65 degrees because I hadn't noticed that it had been turned down the afternoon before trying to cool the house down faster.
That's maybe not 100% true for more modern multi-stage systems that control for humidity, etc., but for most systems it is.
