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Not sure exactly when this happened, but 1.6.1 and 1.5.6 stopped refreshing or adding cities for me. I've tried deleting, reinstalling, blowing away preferences, etc. but nothing seems to bring it back.
Bring up the Console app. Start Meteo. Look for any messages that might indicate what is going wrong. If you scroll down the left side in Console, there is a specific filter for Meteo. That may make it easier to limit what you see. I'm not on my Mac at the moment, but there is also a preference in Meteo to enable extended message logging. Good for debugging but annoying after it's running smooth.