FCE 4.0.1 will Log and Transfer standard AVCHD footage.
If the recorded footage is p (Progressive) it won't work.
Set the camera to record in i (interlaced ) and try again.
Sometimes mixed media on the memory/card can prevent transfer so make sure you test with one type of footage.
If the above does not help provide details of the footage, eg. frame rate, dimension, type.
Also the FCE Sequence Properties in current use.
I also have a few other questions if you don't mind taking some time to see if you could help me:
Are you doing File > Import or are you doing Log & Transfer? File > Import will not work with .mts files.
If you are trying to ingest AVCHD footage directly from a memory card (either via the camcorder or a card reader) you need to use Log & Transfer.
If you only have 'bare' .mts files, FCE cannot import them (the entire AVCHD folder structure needs to be available for FCE to Log & Transfer the video). You would need Clipwrap or similar software to convert the .mts files to QuickTime/Apple Intermediate Codec before importing into FCE.
Note also, that the only progressive mode that FCE works with is 720p, and it does not work with 24fps video at all.