As others have said, if the unit is properly sized for the house, and the insulation is good enough to keep the heat out, and there is enough airflow in the house, the A/C should be able to handle almost any temperature you set it to.
There is a point where it gets so hot outside that air conditioners simply can't do anything to get rid of the heat (the outdoor coil temp equals the outdoor air temp). This is extreme though, >110 deg F. You're getting pretty hot, but shouldn't be at that point yet. (Most A/C units will have specs on this. You can try looking up some specs for it online.)
You indicated the system is turning on and off, so I don't think it's your thermostat.
How cold is the air coming out of your vents? If the system is working properly, it should be as cold as your fridge. If it isn't cold (not cool, cold), you have a problem.
Look at the lines on the outdoor unit. Are they frosting up at all? Is there excessive dripping from your indoor air handler? If your lines are freezing up, turn the system off and call a service company. That means you are probably low on coolant, and running the system without will burn up the compressor.
Other problems that would prevent cold air would be crud on the coil, or the fan on the outdoor unit not working properly.
If the airflow is weak, then you have an airflow problem. Check the air filters in the system. Make sure your ducts aren't plugged somehow. Check for closed dampers or closed vents. Make sure a duct didn't pop loose somewhere (blowing air into the attic instead of out the vents). Make sure your air handler is blowing air.