This has been covered on Stack Overflow: 416 Requested Range Not Satisfiable
Check the error log for the 416 error. [I know you said you checked. It should be there unless you are including a resource from another server.] That should tell you what file. Then make sure it is getting cached correctly [or not cached at all].
What you can do with APC, is disable caching of certain files by using the
apc.filters setting in your
apc.inc or however you are including the apc config.] If you can figure out which file[s] are throwing the 416 error, you can start by configuring APC to not cache them. Sounds like a reasonable start to me.
Actually, come to think of it, if you can find the file throwing the error, you can use APC to, 1, see if it is cached; and 2, check on the file size that APC thinks it is; then 3, check the actual size on disk. Check out this link: devzone.zend.com/article/12618 [the official APC docs kinda suck]