Logic will tell you it can't be the battery when it starts and runs. That should not stop a radio but I think our radios are more computer than radio. Computers are very sensitive to power. I'm pretty sure I've read on this very forum people having all kinds of strange problems because of a dying battery.
Why not just take it to a battery place (I would not recommend Autozone or whatever) and get it checked at least? Most do that for free and if they use a real load tester you will know for sure. They will check the alternator at the same time too.
A sure sign of a dying battery is you will notice after the truck sits overnight (or longer) it just doesn't crank as fast as it used to, that is the battery voltage being too low and it is dying. When you drive it, it will charge back up very quickly letting it start again until that day when it just doesn't. ;(
With all that being said it very well could be something wrong with the radio system but at least the battery is an easy thing to check.