I believe this is a myth, and that 2 flag carriers are permitted from the USA. United's application for SFO-HKG-SGN specifically references this fact in bullet point #2, seen here. It states "the U.S. and Vietnam entered into a bilateral air services agreement ("ASA") in October 2003 under which two U.S. carriers may commence scheduled combination operations between the U.S. and Vietnam." Furthermore, United had to negotiate with Vietnam to accelerate the implementation of 5th-freedom rights from HKG.

My understanding of the situation is that there are issues with the 5th-freedom rights for NW to be able to specifically fly Vietnam from Japan (and even more specifically. NRT). This fact is directly referenced by NWA's response to the Vietnam Airlines (VN) application for SGN-KIX-LAX, seen here. VN requested full open skies authority to carry local traffic on all segments ex-KIX, but NWA's response notes that this violates the existing treaty. NWA further states that they are not opposed to VN's application and recommends full removal of "Annex IV" (which restricts the 5th-freedoms ex-Japan). Presumably, NWA included this so that they may be able to initiate their own services to Vietnam via NRT or KIX in the future.

VN has since answered NWA's response, requesting that they be granted full open skies authority on the route, but with acknowledgment that they would not be permitted to carry 5th-freedom Japan-USA traffic until such time that the overall USA-Vietnam bilateral is altered in regards to Japan traffic. I've heard rumor that VN has already announced a tentative Oct. 27, 2008 start date for LAX, so presumably they anticipate being able to initiate service by then either via KIX without 5th-freedom or via some other point (TPE, perhaps). Either of these would not help NWA in obtaining the Japan-Vietnam operating rights they want/need.