I just spoke with an AppleCare customer service agent today who finally confirmed that I will be getting a replacement for my broken Time Capsule. Needless to say, I'm pretty happy about that. :)
Looking around the net, there seem to be a lot of different stories from people about having their dead TCs replaced. Some seem to not have that much trouble, while others just give up eventually and have their units fixed by an unauthorized service provider. My advice is to get in touch with a customer service agent. If you don't have AppleCare and your Time Capsule is out of warranty, ask the reseller where you got the Time Capsule from who you can get in touch to get a replacement request. The reseller I bought my Time Capsule from was pretty helpful. Apart from me calling up Apple's customer service, one of the supervisors was getting in touch with them as well to help work the issues out (I purchased my Time Capsule from Senco Link Technologies. I purchased my MacBook Pro from them as well and am pretty happy with their after-sales support.)
So now, my concern is whether this replacement that I will get is going to last longer or if it's going to go the same route as my first Time Capsule. I guess I'll just have to wait and see (either in another 17 months, or if and when Apple does acknowledge this issue and sheds some light on what's causing these failures). In the meantime, I think I'll be getting an external HD and will just hook it up to the Time Capsule and back up to that one instead. I know it defeats the purpose of having a Time Capsule in the first place, but I might as well be on the safe side.