Usually happens when you're getting a multimedia text or your WiFi signal isn't too strong to make the phone 'think' it's got a good connection. However, sometimes this also seems to get 'stuck'. Turning Airplane mode on/off should take care of it. (And make sure that the option to boost downloads isn't turned on, or that might also trigger both data streams to be active at the same time when downloading 'big' files).
Just remember, unless you're on WiFi - Calling, MMS will always have to use mobile data, so you can't escape that bit.