There is something wrong. Is there anything else on the card that wasn't recorded by the camera? Or did you shoot different formats on the same card? You might be able to get to the video by digging out the files.
First, copy the complete card content onto a folder on your computer. Inside the folder should be a structure something like this.

Every manufacturer's is different, but it should have an AVCHD bundle. Right-click on the bundle and use Show Package Contents. There may be another bundle inside it like BDMV or some other name. Open that bundle with show package contents. You'll get a folder structure that might have a folder called STREAM, and inside it clips that look like this

Again every manufacturer uses a different layout, and you'll have different content displays, but you should find the .mts files. Select just the .mts files and drag them into the FCP browser. Select the files and right-click for transcode media. Optimize the media.
See if this works for you.