Is it possible that stage 4 breast cancer go undetected on a mammogram?

Yes. Mammography is about 80% sensitive for detecting cancer, so it won't find 20% of them. No test is 100% accurate. So, unfortunately the answer to your question is yes.
No diagnostic test. Is 100% accurate. Stage IV could mean that cancer spread to other organs before it grew big in the breast. So, unfortunately, the answer is yes, certain breast cancers can go undetected on mammogram at any stage.