Finally heard back from my host and am very happy to report there really is no storage cap. Yay!
They also directed me to a section on their legal page that specifically defines their "Unlimited Policy". Simply put, I need never worry about disk storage or network transfers. If, however, VoD ever starts causing problems for other sites on the network, then I may be asked to sign up for a VPS.
I still don't know what caused the attachment error but I can only hope it doesn't happen again anytime soon...