mencoder <input-file> -oac copy -ovc copy -ffourcc XVID -o <output-file>
-oacstands for "output audio codec"
-ovcstands for "output video codec"
-ffourccis for forcing the changing of the fourcc setting. The non-playable files had a FourCC setting of FMP4. I changed it to XVID, which made them playable.
-ois for output file
mencoder The.Big.Show.S01E03.avi -oac copy -ovc copy -ffourcc XVID -o The.Big.Show.S01E03-XVID.avi