There was a time when it was common to carry a cell phone and a pager at the same time. And a Palm Pilot. That definitely goes against the fashion grain, and brings on a ton of functional overlap.
But then there's also that saying, “A man with one watch always knows what time it is. A man with two watches is never sure.”
The real answer is: You do you. Nobody will actively shun you for doing it.
Is it weird to wear both? People don't carry their paintings along, right? Am I lost, is this stupid? Anybody else doing this?