I am updating my entry of March 8, 2012.
After posting my "Solution to Time Machine Error 111," I discovered, as have others, that the fix was not very long-lasting. In fact, I suspect that my "solution" was coincidental. The reason I suspect this is that after a few more weeks, my Time Capsule finally gave an error message that said, "Disk is failing." After about a week or so, the error message changed to, "Disk has failed."
I then got a new Time Capsule (2 TB - grumble, grumble, grumble). I set it up without difficulty and all of my computers are now backing up regularly and have been doing so for over a week as I write this.
So, I have come to conclude that if your perfectly fine Time Capsule starts to fail to back up, even without further error messages, it is going to continue to deteriorate. Once you start to get regular error messages related to the Time Capsule, your hard drive is just about ready to fail, and you should take action to replace your Time Capsule.
I am now on my third Time Capsule. Each of the previous two have lasted about 2 years. You are going to have to consider the annualized cost of the Time Capsule purchase. Others have noted about a 2 year life span for their Time Capsules. I think this is a rather short mean time between failures for this type of device, but that is a personal opinion and based on personal experience and a very cursory look at other postings. The postings were based on problems, not successes, so I recognize I probably have a biased set of data. It is possible that the mean time between failure in a Time Capsule is far longer.