Try converting the video files to another format, such as AIC (Apple Intermediate Codec) or H.264. The free program MPEG Streamclip can convert many formats.
iMovie actually converts the highly compressed AVCHD clips from HD cameras to AIC on import, thereby creating clips that can be edited to the individual frame level - so that's probably the best format to work in. As far as I know, you shouldn't see any degradation in quality when converting to AIC.
Another solution that AppleMan1958 often recommends. Try opening the files in QuickTime 7. If they open, check that there are no other tracks other than video and audio (soundtrack). Other tracks, such as Chapters, Text etc should be deleted - iMovie won't work with these extraneous tracks.
To check the tracks, in QuickTime 7's Menu go to Window>Show Movie Properties. You will see the individual tracks listed in the window that opens, with tick marks against the active tracks. Click on the unwanted track or tracks, then click on the delete button in the top left corner. Before doing any of this, it's best to copy the files beforehand, just in case you delete something you shouldn't.
After saving the files, hopefully they will now import to iMovie. If not, try the conversion method I described earlier.
