It is not completely true. Babies develop the lacrimal system at 42 days, in utero. The canals and ducts for tears are formed at 60 days .
At birth, tear production by the lacrimal gland is minimal. Normal tearing develops several days to 2 weeks after birth.
Some 6% of newborns are born with a tear duct obstruction (in some online articles it stated up to 10%, but I think my source is more reliable). The child still produces tears, but they can well up on the eyelid .
Other more serious abnormalities can cause this problem. Duct cysts or hemangiomas can also cause tear duct blockages in more severe cases. Various factors, such as anatomical abnormalities and problems with pumping tears increase the propensity for the disorder in children with Trisomy-21 . As nico has alluded to in the comments, there may be a lag in the development of the physiology of tear production even in non-Trisomy-21 children.
 Thomas K. McInerny, MD, FAAP, ed. 2009. American Academy of Pediatrics Textbook of Pediatric Care. Elk Grove Village, IL. American Academy of Pediatrics, p1706.
M. E. Wilson et al. (eds.), Pediatric Ophthalmology, 2009, pp. 276-284. DOI 10.1007/978-3-540-68632-3_1.