In addition to being a freedom-loving zealot :-), I'm a mechanical engineer. So I know a bit about thermodynamics and air conditioning.

The A/C is not "set" to provide a certain temperature difference, but there is a maximum heat-transfer capacity to the system. The AC works by cooling a coil inside your furnace plenum (the evaporator coil), and then the furnace fan blows air through the evaporator, cooling it. Heat from the air blown across the evaporator is transferred to the refrigerant, which flows outside to the condenser coil (the one outside), where it is exhausted to the outside air.

On really hot days, the air inside the house can be theoretically heated by the ambient environment faster than it can be cooled by the evaporator. As a result, the inside air temperature will reach an equilibrium higher than the thermostatically set temperature (though lower than the outside temperature). In practice, though, you should be able to get it down to 72 degrees as long as you're not repeatedly opening the windows at night to let in a bunch of humidity (you should generally turn on the AC and leave it on for a prolonged period, to avoid repeatedly drying the inside air -- this is referred to as "latent heat of vaporization". Opening with windows at night is the worst thing you can do in a humid climate).

On these days, then, it is really important that you are getting as much capacity from your system as possible. The two major things that decrease capacity are dirty/clogged coils (which reduce airflow across the coils) and low refrigerant level.

What you are describing is usually indicative of reduced performance due to a frozen evaporator. Water vapor condenses out of the air as it passes by the cold evaporator. If the evaporator is clogged or dirty, or if the condensate drain line is clogged, the water will freeze in the evaporator. Once this starts to happen, it reduces the airflow across the evaporator, causing more freezing, until the evaporator is totally frozen and clogged with ice. If you go look at your plenum, you should see two copper lines heading into it and my guess is that one will be completely ice covered. If that's the case, you need to turn off the system (switch the fan to on), and let it thaw out and dry. You can limp through these hot days by thawing out the evaporator every 5-6 hours, but to fix it you'll need to figure out why it's freezing up and resolve that. Cleaning the evaporator is a really difficult task. Unplugging the condensate drain is fairly easy.

Go check your evaporator and if it's frozen up we can talk more about that. If your evaporator is bone dry and cold, then you either have a capacity issue (AC can't keep up), or you may be low on refrigerant due to a system leak.