This also happens with pre-made ice.
I have a separate ice machine - you know, one of those portable ones that you add water to yourself. I usually make and bag some ice ahead of time for those times when I want iced water, tea, coffee, etc..
And if the ice is stored in the freezer two long, it begins to pick up smells from the freezer right through the bag!
So yes, ice CAN go stale and smelly even when it's bagged, much less in your in-freezer unit's storage bin. And it doesn't taste very good either.
I also agree - dump it and make some new ice cubes.