Thank you for the reply, sanjose-san
I had tried to open the folders to find no issues. No wonder, since backing-up had been working fine by then, too.
I logged off/on, and rebooted, and now I can open Time Machine.app. I don't know which action has fixed the problem.
So, as for the MacPro and back up to an internal HDD, the problem seems to have been resolved.
I, however, got yet another Time Machine related issue for MacBook Air and an external HDD. This time, the backing up is so slow, and would never finish. After upgrading to 10.7.5, 4 GB was to be backed-up, and of which only 1.5 GB were backed up after three hours. Then, I initialize the external HDD, and started over the back-up. At very beginning, it seemed working fine, but it slowed down after 4GB and now to have stalled altogether at 4.8 GB (36.2 GB to be backed up in all.) The access lump of the HDD suggests that the access to the disk is very rare and intermittent.
Is this a feature or something we can improve?