Question: I've been regularly backing up my 2011 MacBook Pro using Time Machine with two separate external drives (1Tb at work, 2Tb at home) while running Mavericks. 

Today I upgraded to Yosemite, and Time Machine seems to think I have a brand new computer - the backup on the 1Tb disk failed with "not enough space", and the backup at home is currently attempting to back up the entire 350Gb on my computer.

(Previously with the 1Tb drive, Time Machine would automatically delete old archives when the drive was full.)

Is this a common problem (ie a bug)? Is there a way to tell Time Machine to continue where the last backup finished, rather than starting a brand new "branch"?

Answer: 
You may want to check this thread: <https://discussions.apple.com/message/27031297>

It seems that if you backup external drives with Time Machine and you disconnect them at a later point, for instance for a few hours or days, and reconnect them, Time Machine will want to make a full backup of them (even if nothing has changed). That behavior appeared with Yosemite. I have not heard about any solution or workaround (yet).
There's some coding in Terminal you can do to remove flags osx or os sierra applies to those TM backups, then retry TM again. You can try repairing the sparsebundle files in recovery mode (backup current iteration in addition to the SB to an HDD to be safe).