This would have to depend on the person. Some can survive more than a week without having to pass bowel. The danger in this is that, this can lead to further complications or even to sepsis, which is a general and widespread infection within the body. Consult your doctor if you have an absence of bowel movements since this may be caused by a more serious condition such as a blockage.
Depending on what medications you are taking, or if you have had any recent surgeries, you may not have a bowel movement for a week. You should always check with your doctor about it though. Go here to read more: http://gutsense.org/gutsense/constipation.html.