The problem is usually that someone is pulling far more than they are actually able to deal with, which puts an incredible amount of stress on the healer, and can quite easily lead to someone (or everyone) dying if the healer isn't playing perfectly. In the (likely) event that everyone dies, suddenly it's the healer's fault for not keeping everyone alive, and no healer wants to deal with that heaped on top of the strain of doing everything they could to heal through it.
EDIT: Ooh, I have another story; I forgot to post this as my static grabbed me immediately after, but the GIFs thread just reminded me;
One of my runs in A12, I was playing as a PLD and MT'ing the fight. Later on in the fight, there's been a couple of deaths, and the healers are a bit strained, with weakness as they had both died previously. I notice that both of them are stacked up - alone - in the Shared Sentence mark with no one else nearby. I have the Aggravated Assault debuff (and thus would usually avoid the Shared Sentence)... but in a split second, I knew that we would not recover if I didn't stack too, so I pop hallowed and ran in at the last moment. The healers barely survived, and we managed to clear without wiping.
I got a /hug out of one of them afterwards. =^_^=
