Why were my appliances and air conditioner failing to start for a short period?
Late in the afternoon on a hot day is when the power company has the highest load. Later in the evening the load tapers off.

So, either your air conditioner is malfunctioning and drawing way too much current, or the power company was experiencing a brown out, or a little of both.

If you plug the A/C back in and it works fine then maybe it was the power company.

If you continue to have problems I would suspect the A/C is going bad. The only way to tell is to take ammeter readings on the unit while it is running to see if it is exceeding its rating.

There are other possibilities but if you don't experience any problems without the A/C running I wouldn't pursue them.

Good luck!