**Yes—78°F indoors at 105–106°F outdoors is achievable with a properly sized, functioning central AC. But whether your particular system should maintain it depends on your local climate, the house’s heat gain, and the system’s capacity.** Reaching 89°F with a 78°F setting is enough of a shortfall that I would investigate it.

AC systems are sized for a location’s *design temperature*, rather than its hottest possible day. If 106°F is well above that temperature where you live, a healthy system can run continuously and still fall behind. If temperatures like this are routine locally, 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 is **no universal “20°F below outdoors” limit**. The often-mentioned 20°F difference refers roughly to the temperature drop between indoor air entering the AC and cooled air leaving it. That air recirculates, so 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/)

Your second day being worse despite a slightly lower outdoor high could reflect heat retained in the walls, roof, and furnishings, especially after the warmer night. That is a plausible explanation, not a diagnosis; humidity, airflow, or equipment trouble could also contribute.

A few useful checks:

- **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. An indoor fan running does not establish that the compressor is working.
- **Measure return-air and supply-air temperatures** with an air thermometer while cooling is running steadily. A drop around 20°F is a useful rough check, but humidity and airflow affect it, and a normal drop does not prove adequate overall capacity.
- **Look for visible ice** on refrigerant tubing or the indoor coil. If present, 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/)

Given the **11°F gap from your setting**, I’d arrange an HVAC check, especially if the system previously handled similar weather. Ask them to verify 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 recover to 78°F overnight? Those details would help distinguish extreme-weather limits from a performance problem.