Time machine back up failed, why?

+1 Any of the above reasons. If your external HD is 4-6 yearly old, it is not a bad idea to consider replacing it. All HDs die, it is not a matter of if but when. So to be safe replacing them proactively is wise. If it still works okay, just repurpose it for some other task that is not critical like Time Machine.

There could be a myriad of reasons. Reformat the drive, run First Aid and try again.

If the drive is several years old, it may be failing and in need of replacement.