It is your OS that is the problem.. Let me guess.. hmm.. so hard.. Yosemite!!
welcome to the dog house..
If you expect Time Machine (software in the computer to backup) to run reliably do not turn it off.. it must stay on and continually backup..
Otherwise buy a backup software like Carbon Copy Cloner or Chronosync.
You can get TM working again.. but first of all do the setup of the Airport Extreme again.. Follow all the following rules.
Factory reset universal
Power off the AE.. ie pull the power cord or power off at the wall.. wait 10sec.. hold in the reset button.. be gentle.. power on again still holding in reset.. and keep holding it in for another 10sec. You may need some help as it is hard to both hold in reset and apply power. It will show success by rapidly blinking the front led. Release the reset.. and wait a couple of min for the AE to reset and come back with factory settings. If the front LED doesn’t blink rapidly you missed it and simply try again. The reset is fairly fragile in these.. press it so you feel it just click and no more.. I have seen people bend the lever or even break it. I use a toothpick as tool.
Then redo the setup from the computer with Yosemite.
1. Use very short names.. NOT APPLE RECOMMENDED names. No spaces and pure alphanumerics.
eg AEgen5 and AEwifi for basestation and wireless respectively.
Even better if the issue is more wireless use AE24ghz and AE5ghz with fixed channels as this also seems to help stop the nonsense.
2. Use all passwords that also comply but can be a bit longer. ie 8-20 characters mixed case and numbers.. no non-alphanumerics.
3. Ensure the AE always takes the same IP address.. this is not a problem if you use it in router mode but it will continually change IP if used in bridge.. so change to static IP mode instead.
https://discussions.apple.com/thread/5981989?answerId=25135547022#25135547022&ac_cid=tw123456#
4. Check your share name on the computer is not changing.. make sure it also complies with the above.. short no spaces and pure alphanumeric..
5. Make sure IPv6 is set to link-local only in the computer. For example wireless open the network preferences, wireless and advanced / TCP/IP.. and fix the IPv6. to link-local only.
6.
Mount the TC disk in the computer manually.
In Finder, Go, Connect to server from the top menu,
Type in SMB://10.0.1.1 (or whatever the TC ip is which you have now made static. As a router by default it is 10.0.1.1 and I encourage people to stick with that unless you know what you are doing).
You can use name.. SMB://AEgen6.local where you replace AEgen6 with your TC name.. local is the default domain of the TC and doesn't change.
However names are not so easy as IP address.. nor as reliable. At least not in Yosemite they aren't. The domain can also be an issue if you are not plugged or wireless directly to the TC. Please to be of any consistent use.. the computer that is backing up must connect directly to the AE and only the AE.. by wireless or ethernet doesn't matter.
There is a lot more jiggery pokery you can try but the above is a good start.. if you find it still unreliable.. don't be surprised.
Once you complete the above, redo the setup of the TM..
It will say it is going to do a new backup but will actually find the old one and use it.. however it will take ages over wireless as it must deep scan the old backup due to the week break in backups.
Time Machine needs to run hourly.. if the backup is too slow over wireless use ethernet or even better wipe the USB drive and start over with it plugged directly into the computer.. it is faster and more reliable.
