Yes, there is such a condition where a person could be breathing without having a pulse. Victims who are not breathing but still have a pulse are given so called "rescue breathing". This action works because the air you breathe out as a rescuer still has oxygen in it. Air itself is made up of about 21 % oxygen, but when you breathe it in, your body uses a small part of it. Therefore, the air you breathe out and into a victim is made up of about 16 % oxygen. That is enough to keep someone alive.