This looks like a duplicate question:
416 Requested Range Not Satisfiable
check the explanation here:
But basically, 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 certyain files by using the apc.filters setting in your php.ini [or 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.
AAAACtually, 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 & 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]