"At that time" = at that period in time. Eg, "She was a movie star in the 1950s. She was very famous at that time. Today, however, no one has heard of her." "That time" = that one time. Usually, it refers to an exception to a rule. "Joe is usually a jerk. But he helped me after my mother died; he acted very nicely that time."

'That time' in the context referred to above is referring to a period of time over a few hours. Or a period of time in contrast to another period of time when the person did not perform well. "At that time' is more about a moment in time. At least in the way you have referred to above.