what may have happened to my connection? Is it possible that the router is malfunctioning?
Yes, the TC can experience hardware failure at around the 3 year mark.
Yours is a gen3 I would assume.. look at the bottom rubber mat and check for A1xxx model.
It should be Gen3 A1355 or Gen4 A1409
So the steps to take. Subsequent to the ones you have already done.
1. Full factory reset.
Universal Factory Reset.. any model TC or AE.
Unplug your TC/AE from power or turn off at the power point.
Hold in reset. and power the TC/AE back on.. all without releasing reset and keep holding in for about 10sec. (this is often difficult without a 2nd person or a 3rd arm).
Release it when the status light flashes rapidly. If it doesn’t flash rapidly you have missed it and try again.
Note..
Be Gentle! Feel the switch click on. It has a positive feel.. add no more pressure after that.
TC/AE will reboot after a couple of minutes with default factory settings and will wipe out previous configurations of the router.
No files are deleted on the hard disk in a TC.. No reset of the TC deletes files.. to do that you use erase from the airport utility.
2. If the firmware is 7.6.4 which is the latest.. it is fairly buggy.. take it back to 7.6.1 and on a Gen3 take it back to 7.5.2 .. it is easy to upgrade again..
Downgrade by holding the option key when you click version on the summary page.
3. New clean setup.. ensure all the things I recommended.. plus different names on the two bands so you can see which is having problems.
4. Use a wireless diagnostic tool to see if you can see some issue with the wireless signal.. It is built into Mountain Lion and Mavericks.
See http://support.apple.com/kb/HT5606
No luck I would say the TC is simply grown old and tired.. and in need of replacement.