One more interesting note. If I rename the files, they will import on the next attempt.
How did those MTS files get into your computer in the first place? And are they are on your C or main drive or on a second or external drive?
I always recommend opening a Premiere Elements project using the proper settings for that camcorder and then using the Get Media tool to open the capture tool or Video Importer.
This often activates components for working with and viewing that video that aren't otherwise activated.
And, of course, make sure you also have the latest version of Quicktime, per the program's requirements.
Did you get any message while you import these mts files in the Organizer ? Does it happen for other file types also ?
Also, can you check if Video is checked in View->Media Types->Videos. Please also check that Hidden file filter is not checked in Edit->Visibility
On reimporting, if no new files are being added, this means that those files are definitely in your Organizer. There is just some setting which is not allowing you to see those files.
As SG)PSE said, you may want to check your settings.