Do those who have down syndrome always have the physical characteristics to show it?

Depends...: I would say it would be very unlikely for someone with "regular" Down syndrome (extra chromosome 21 in all cells of the body) to go undiagnosed. Mosaic Down syndrome (where not every cell in your body has an extra chromosome 21), however, is quite variable and can be "mild" enough such that someone may have subtle physical characteristics as to be not be diagnosed.

No: The vast majority of ds patients will have enough similarities in physical appearance to be recognised by knowledgable people.Yet all do not. I recall a case where an infant appeared to have many features of a different chromosome defect had 21 trisomy, verified by 3 separate studies.I have also seen a case thet had features so mild, the 21 trisomy was initially considered an error.