It is tough to decide who is responsible. Bed bugs can come from anywhere - they can come in on animals or in infested furniture, they can come in the home from outside. There is no way to prove whether these little guys came from you (such as if you bought used furniture that had some you didn't notice), someone else's apartment, or just came in from outside.
Rather than pointing fingers at who's to blame, I would focus on getting rid of the critters. Have the landlord come in and exterminate all of the apartments. He needs to do it NOW rather than waiting for you to move, because the longer he waits the more chance there is of the little guys moving from your apartment to someone else's.
Wikipedia has some ways that you can kill any of the ones you see, including spraying rubbing alcohol on them or using a fabric steamer. Short of that, I would say toss the mattress, check for bedbugs in the rest of your furniture and either kill them or toss the furniture, and then go ahead and move. Let the landlord take care of getting them out of the apartment. However since bedbugs feast on human blood, they will more than likely migrate once you leave the apartment.