Personally, I don't think there is much of a case here. There hasn't been any trade or service mark infringement. Neither party is trying to mislead the other to the wrong site. The parties are in unrelated businesses, so the actual overlap of common customers is likely to be very small. (IANAL; usual disclaimers apply.)
If anything, this is similar to the "slashdot" effect of a link being placed on a very popular site (with users who tend to follow links) to a less popular site (not necessarily equipped to handle a sharp traffic increase). Perhaps UTube has a case against the rude folks who are disrupting customer service, but otherwise, I think they just need to ensure that they have sufficient provisioning (for their entire business, not just bandwidth). These things happen on the net; they are part of the cost of doing business.