There are a few causes..
Using the More Like This..
the top reference is helpful.. but you can see this is a regular problem.
Basically reboot the whole network.. restart in correct order.. modem.. router/TC...Clients.. 2min gap
Half the time will fix it.
It can also be the backup is corrupt and needs repair.
Are you aware of the supplement to 10.7.5 to fix TM??
This is not related to your problem,, rather it is slow backups.
I would like to make a comment. When I had the same error, it occurred to me that I resently changed my password on the Time Capsule. After changing the password back it worked again. So:
Problem: Error code -6584 (Fejlkode -6584). Changed password on Time Capsule
Solution: Changed it back
Good luck Dan