According to High-Def Digest, Paramount stated that "the bandwidth capacity was increased in preparation for the release but the demand exceeded all expectations so capacity was expanded dramatically last night and local servers were established worldwide to accommodate all the fans".
Ouch... Looks like someone underestimated the traffic! Scalability issues are tough: permanently sizing your systems for peak traffic is expensive, but so is customer frustration! You have to expect that everyone will try to access online content on release date. If you're trying to serve them all from a single location, that could mean a lot of trouble.
Not sure how they solved it. The PR seems to imply that they replicated the content to "local sites" to spread the load (which is AFAIK what iTunes also does).
This is not free, far from it. It will be interesting to see how Paramount handle this in the long run : in-house, CDN, 3rd party, ...