There are batteries inside the UPS that need to be replaced periodically. I have one that does the same thing so I stashed it in the back of a closet. If I ever decide to get it out to use again I'll have to change out the rechargable batteries. (What brand is yours?)
I think it's like the truck battery--it will run the lights but doesn't have enough juice for the starter. That's when the automotive battery gets replaced, not when it's completely dead. Same thing for yours.
I don't think they're cheap. You might have to buy a new UPS and get those new batteries. It's a waste, I know--I'd rather not toss an entire device because of the expensive batteries (it happens with cordless phones all of the time--the phone new costs $19 on sale, but a replacement battery costs $50).
I would first check with some of the aftermarket battery places to see what you can come up with. Another analogy is like the printer/toner situation. Companies that sell printers at cost are actually in the business of selling ink and toner. If you believe their rhetoric, you'll buy the name-brand expensive ink to keep your printer in warranty. But you don't have to, you can find good after market brands for a fraction of the price. So with this pep talk on the habit of batteries and the perfidy of the marketplace, go forth and find a replacement rechargeable battery (then tell the rest of us where you found it!)