Q:

Is it possible to buy a ticket where the luggage cannot be checked to the final destination? 

My brother recently flew on a flight from the USA to Africa involving three flights on three different airlines.  When he checked in, the agent told him that it was impossible to check the bag all the way to the final destination as they did not have an interline agreement with the final airline.  As a result, he was forced to pass through immigration/customs on arrival in South Africa to recheck the bag, missed the last flight of the day to Mozambique and spent the night at an airport hotel in Johannesburg (at the airline's expense). 
I have a hard time believing that they could sell a ticket for an itinerary that does not allow the bag to be checked to the final destination.  Was the gate agent correct or just incompetent?  The itinerary was booked as a single ticket through one of the major online travel websites.  Also, some of flights were codeshares, but I'm not sure if that makes any difference for luggage. 
UPDATE: I know that on international to domestic connections it is usually necessary to clear immigration/customs and recheck the luggage.  The same is true for an international to international connection in the USA, but in both of these cases all passengers are required recheck bags even though the luggage should be tagged to the final destination.  I'm asking about a situation where I know that is possible to change planes without rechecking luggage, but for some reason depending on the ticket or airlines involved they will not check the bag through. 

A:

Yes, I have seen this on many occasions, on flights to USA, Europe and Australia with multiple connections.  
The scenario is exactly as you describe - at the airport where they cannot be checked through, you need to pick up your luggage then check it back in for the next leg of your journey.  A long as you plan for it, it ends up just being a mildly irritating time lag. 
My guess is that it happens when the different airlines' systems aren't well integrated so cannot pass flight information through. 

A:

Yes, it is an there are many reasons why it can happen.  The airlines not communicating is one of them but I think the most common reasons are policies of transit airports.  This is why luggage is nearly always tagged to its destination with all flights and yet the passenger has to move it himself.  In some airports there is an express way to check luggage back in but if they force you to pass immigration, it can take longer. 
The international to domestic transition and vice-versa has already been mentioned and is particularly related to the airport policy.  Since it can imply a terminal change, it can make logistics complicated.  Airports may decide for any reason, including local and global laws and regulations, that they simply require passengers to collect their luggage and check it back it.  This allows local people to validate, inspect and approve (or not) the luggage in question. 
The best way to diminish the issue is to contact someone.  The airline should know the policy based on which airport and terminal your flights are scheduled for.  With this information you can decide to go through another route which I have done many times before or choose flights with longer transit times, depending on what is available.  Just note that different regulations may apply going and back, so make sure to check both halves of the trip. 

A:

Without knowing which airlines were involved, I assume that noone is able to answer your question reliably, but it might very well be that the luggage really couldn't be checked through to the final destination.  
Another common scenario for this is if the last leg of a multi-leg trip is a domestic flight.  In this case, customs must be cleared at the first airport in the destination country.  Unless there are any local facilities to make the process easier, this means that you have to go through immigration, pickup your luggage, clear customs, recheck your luggage and go through security before you can board your last flight.