Ok, thanks for the update on the modem. If you got your Time Capsule (TC) new in 2012, then it most likely is a 4th generation 802.11n "flat" model. The latest versions are "tall." Regardless, it's important to understand that most consumer-grade networking hardware have a useful lifespan of between 3-5 years. I'm not necessarily suggesting that your TC is faulty, just making you aware of that bit of information.
What I would suggest at this point in time is to perform a "factory default" reset on your TC, and then, set it back up for your networking goals. Note that performing this type of reset does NOT affect any data stored on its internal (or external) hard drive(s).
After you have it back up and running, see if Time Machine continues with its backups.