Which Garmin device do you have? Are you seeing a message which indicates that there's no room for data? Make sure you have the latest version of Garmin Connect Mobile iOS, which just came out yesterday.
If you need to delete data from your device, typically you'd delete activities from history, or via USB cable on a Windows computer delete the fit files located in the \Garmin\Activity folder. However, I would make sure you've synced using Garmin Express, before you've deleted everything.
