A properly sized, functioning central AC can keep a house at 78°F when it’s 105 to 106°F outside. Whether yours should manage that depends on your local climate, how much heat enters the house, and the system’s capacity. With the house reaching 89°F, an 11°F gap from your setting, I’d arrange an HVAC check, especially if it previously handled similar weather.

AC systems are sized for a location’s design temperature, which can be lower than the hottest temperature the area experiences. If 106°F is well above your local design temperature, a healthy system may run continuously and still fall behind. If that heat is routine where you live, maintaining 78°F is a more reasonable expectation. [ENERGY STAR’s design guidance](https://www.energystar.gov/partner-resources/residential_new/working/hvac/hvac_designers/design_temp_limits)

There’s no universal limit of “20°F below outdoors.” The commonly mentioned 20°F difference is roughly the temperature drop between indoor air entering the AC and cooled air leaving it. Because that air recirculates, the house can become much more than 20°F cooler than outside. [Carrier’s explanation](https://www.carrier.com/us/en/history/frequently-asked-questions/)

The worse performance on the second day, despite a slightly lower outdoor high, could come from heat retained in the walls, roof, and furnishings, especially after the warmer night. That’s one possible explanation; humidity, airflow, or equipment trouble could also contribute.

You can check a few things:

- Check the filter and make sure supply vents and return grilles are open and unobstructed.
- Observe the outdoor unit while the thermostat calls for cooling. The indoor fan can run even when the compressor isn’t working.
- Use an air thermometer to measure return-air and supply-air temperatures while cooling runs steadily. A drop around 20°F is a useful rough check, though humidity and airflow affect it. A normal drop doesn’t prove the system has enough capacity to cool the house.
- Look for visible ice on the refrigerant tubing or indoor coil. If you see any, turn cooling off and arrange service.

Dirty filters or coils, refrigerant problems, and insufficient capacity can all cause this behavior. [Carrier’s troubleshooting guidance](https://www.carrier.com/us/en/residential/hvac-resources/air-conditioners/ac-not-blowing-cold-air/) Ask the technician to check airflow, refrigerant charge, coil condition, and duct losses before concluding that you need a larger unit.

What city or region are you in, and does the AC get the house back to 78°F overnight? Those details would help distinguish the system’s limits in extreme weather from a performance problem.