Hmmmm, well it very likely could be some global hook called by a product that would affect the inlinemod.php file I suppose. Have you tried disabling ALL your products for a day and seeing if this still happens?
Ok, I disabled all but Downloads II, Multiple Login Detector, and 2 custom mods. 4 days, no problem. If the problem recurs, I could disable the others, but that would be problematic for our site. I guess if it turns out to be a specific mod, I will have to live with it and check the load daily, and check the backups.