The reason for my suggestion to browse the backup on the copied drive is that when one of my machines seemed to want to start from scratch with a new backup, I browsed the backup and TM "found" the backup again.
At the time I was backing up an iMac computer to a drive connected to my Mac Pro. I believe I had disconnected the drive physically from the Mac Pro, and then reconnected it. TM on the iMac lost the connection, and when I reconnected to the drive to the iMac over the network, TM on the iMac didn't see it as the backup drive anymore. That is, there was no reference to oldest, last or next backups at all, and it looked like it was going to create a new backup. Worried that I was going to have to recreate the full backup (a very slow process), I tried browsing the backup using TM. When I went back to the TM systems preferences panel on the iMac, the oldest, last and next backups had been restored, and it did the next backup as expected.