The number of apps that run in the background are very few and far between.
When you say only the Messages and Music apps are open, does this mean you are listening to music with the iPhone's iPod while exchanging iMessages meaning both apps are running?
Unless an app is running in the background, removing an app from the recently used apps bar or dock that is not running in the background removes the app from the recently used apps bar or dock only.
What is your cellular reception at the time?
Is the iPhone connected to an available wi-fi network at the time?
Is Bluetooth enabled when not being used?
What is the screen brightness?
Are any email accounts being checked for new email automatically in the background with the iPhone's Mail app or are you accessing an email account that supports Push access for received email?