time machine failed backup notification

Not in a normal way.

A really good programmer could probably get into the Mac operating system commands and change the notification timeframe to a different value though.....for a very steep price.

An easier "fix" for some users might be to simply go into Time Machine Preferences and turn off automatic backups by unchecking the box next to Back Up Automatically.

Then, backups would only occur when you back up manually, and you won't see any messages reminding you that Time Machine has not backed up in xx days.