If an image or video is missing metadata date/time fields, then LR 3.4.1 uses the file's date-modified for filtering, sorting, and pre-populating the Metadata > Edit Capture Time dialog, but it uses the file's date-created to display in the grid view. It should use date-modified consistently for all of those. (When Windows copies or restores a file, the date-modified is preserved, but date-created is usually set to "now".)
This problem trips up people managing scanned images and videos, since scanners typically don't add any metadata and LR doesn't understand much video metadata.
This is fixed in Lightroom 5.5, which is now available. (Note that there are several distinct problems discussed in this thread, so when I say that "this is fixed" I'm referring to the problem which John Ellis originally reported.) Thanks, Ben