Several users have been reporting this problem since updating to the latest version of iTunes. Have you had a look at this Apple support document yet related to your issue though just in case?
I think I've found the problem here and its solution:
I use Handbrake 0.9.5 (mostly) to re-encode avi movies (and the like) for transfer to my Classic. When I launch that application, the default translation preset is "Regular/Normal." And therein lies the problem, for "Regular" most certainly does not work on the iPod Classic! It does for my MacBook Pro, to be sure, but NOT for my old iPod.
However, the moment I select "Apple/iPod" as the preferred encoding preset, a check mark immediately appears in the small "iPod 5G Support" box . . . which is NOT checked when "Regular/Normal" is the default. Eureka!
Now I have no problems moving movies to my Classic. Nobody's at fault here . . . I just had to look a little closer at the available options.