Forced air system blowing cold air through bedroom vents, hot through the rest
It definitely sounds like that section of duct has a leak or travels through an un-insulated part of the structure. Sometimes the cavities between walls or under floor joists are used as duct or returns and they are not enclosed. Leaks in the building envelope, especially with such cold weather, could keep that area cold all the time and any warm air blowing through it will get cooled down right away. 

If you are feeling a draft out of that vent when the heat is not blowing, that is concerning and may indicate a large air leak. There are a variety of methods for testing for duct leaks. One common method is by using a blower door on an external door to get a baseline leakage number for the home. Then, all of the vents are covered and another leakage number is obtained. The difference indicates the leakage of just the duct system. A service that does this is usually inexpensive (under $100) and can tell you just how much air the ducts are leaking in additional to helping you find other sources of cold air leaks into your home. 

If the landlord isn't willing to pay for investigating it, you may want to pay for it yourself since you could then bring him/her hard numbers and ultimately save on your heating bill once repairs are made. If the landlord still will not budge, I'd make the claim that warm building air mixing with cold duct or external air could cause mold growth and make you sick.