Old dragons can frighten you:
Frightful Presence. Each creature of the dragon's choice that is within 120 feet of the dragon and aware of it must succeed on a DC 16 Wisdom saving throw or become frightened for 1 minute. A creature can repeat the saving throw at the end of each of its turns, ending the effect on itselfon a success.
Calm Emotion can suppress this:
You can suppress any effect causing a target to be charmed or frightened. When this spell ends, any suppressed effeet resumes, provided that its duration has not expired in the meantime.
Do you keep repeating the saving throw while the frightened condition is suppressed?